#7509538

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,20)(19,23,28,26)(21,24,31,29)(25,27,32,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19,22,28)(18,23,20,26)(21,30,31,27)(24,25,29,32);(9,12,13,11)(10,14,15,16)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,27,26,30);(2,7)(3,8)(10,15)(14,16)(17,20,22,18)(19,26,28,23)(21,29,31,24)(25,30,32,27);(1,4,5,6)(2,3,7,8)(18,20)(23,26)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416734);