#7633180

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#34937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=g,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540376);