#7532933

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,30,23)(20,32,34,24)(22,26,36,44)(28,39,33,42)(29,40,48,47)(31,45,37,41)(35,46,38,43);(2,6)(7,8)(11,16)(13,15)(17,19,21,30)(18,23,25,27)(20,33,34,28)(22,37,36,31)(24,42,32,39)(26,45,44,41)(29,38,48,35)(40,46,47,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,34)(22,35)(23,39)(25,32)(26,43)(27,42)(29,31)(30,33)(36,38)(37,48)(40,41)(44,46)(45,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,29)(21,34)(22,38)(23,40)(25,32)(26,46)(27,47)(28,31)(30,48)(33,37)(35,36)(39,41)(42,45)(43,44);(2,6)(7,8)(20,34)(24,32)(28,33)(29,48)(35,38)(39,42)(40,47)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440129);