#7632449

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=i,e^2=i,f^2=h,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, 7539645);