#507736

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

529 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414932);