#7968075

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4714 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875271);