#510006

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,5,8)(3,4,7,6)(9,10,16,18,13,20,14,21)(11,23,29,41,26,17,27,43)(12,30,35,19,32,48,33,46)(15,22,38,44,36,49,37,50)(24,52,55,42,54,51,28,39)(25,56,59,45,31,61,57,40)(34,60,66,47,64,71,65,69)(53,68,58,70,62,67,63,72);(1,3,5,7)(2,6,8,4)(9,11)(10,17)(12,31)(13,26)(14,27)(15,24)(16,29)(18,43)(19,45)(20,23)(21,41)(22,51)(25,32)(28,37)(30,56)(33,59)(34,62)(35,57)(36,54)(38,55)(39,44)(40,46)(42,50)(47,70)(48,61)(49,52)(53,64)(58,65)(60,68)(63,66)(67,71)(69,72);(1,4,5,6)(2,7,8,3)(10,18)(11,24)(14,16)(15,37)(17,39)(19,30)(20,21)(23,42)(25,53)(26,54)(27,55)(28,29)(31,62)(33,35)(34,66)(36,38)(40,67)(41,52)(43,51)(44,50)(45,68)(46,48)(56,70)(57,58)(59,63)(60,71)(61,72)(64,65);(2,8)(3,7)(9,12)(10,19)(11,25)(13,32)(14,35)(15,34)(16,33)(17,40)(18,30)(20,46)(21,48)(22,47)(23,45)(24,53)(26,31)(27,59)(28,58)(29,57)(36,64)(37,66)(38,65)(39,67)(41,56)(42,68)(43,61)(44,60)(49,69)(50,71)(51,72)(52,70)(54,62)(55,63)

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417202);