#7509539

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	#30584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(17,18,22,25)(19,24,30,32)(20,23,26,21)(27,31,29,28);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,32)(26,31);(9,12,13,11)(10,14,15,16)(18,25)(19,27)(21,23)(24,28)(29,30)(31,32);(2,7)(3,8)(10,15)(14,16)(17,20,22,26)(18,21,25,23)(19,28,30,31)(24,29,32,27);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,26,25,20)(19,29,30,27)(24,31,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=i,e^2=fi,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.

Relationships

maximal subgroup of	#72313287, #72313550, #72313698, …
maximal normal subgroup of	#72313287, #72313550, #72313698, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416735);