#510032

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×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 534 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)(9,10,16,38,35,53,36,18,13,20,14,21,15,22,40,48)(11,23,39,44,28,46,56,69,26,17,27,42,24,45,61,47)(12,29,62,52,33,64,37,19,31,63,32,51,34,54,41,50)(25,57,66,72,55,71,60,49,30,65,58,70,59,68,67,43);(1,3,5,7)(2,6,8,4)(9,11)(10,17)(12,30)(13,26)(14,27)(15,24)(16,39)(18,47)(19,49)(20,23)(21,44)(22,46)(25,31)(28,35)(29,57)(32,66)(33,59)(34,55)(36,56)(37,67)(38,42)(40,61)(41,60)(43,50)(45,53)(48,69)(51,70)(52,72)(54,68)(58,62)(63,65)(64,71);(1,4,5,6)(2,7,8,3)(10,18)(11,24)(14,36)(15,35)(16,40)(17,42)(19,29)(20,48)(21,22)(23,44)(25,55)(26,28)(27,39)(30,59)(32,37)(33,34)(38,53)(41,62)(43,68)(45,69)(46,47)(49,71)(50,63)(51,54)(52,64)(57,70)(60,67)(65,72);(2,8)(3,7)(9,12)(10,19)(11,25)(13,31)(14,37)(15,33)(16,41)(17,43)(18,29)(20,50)(21,54)(22,51)(23,49)(24,55)(26,30)(27,60)(28,59)(32,36)(34,35)(38,64)(39,67)(40,62)(42,68)(44,71)(45,70)(46,72)(47,65)(48,63)(52,53)(56,58)(57,69)(61,66)

Polycyclic presentation:

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

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, 417228);