#8741642

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

4913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8648838);