#7632539

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	#36434	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4940 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=i,e^2=i,f^2=g,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, 7539735);