#509581

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416777);