#8932580

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

4940 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)(10,13)(14,16)(17,18,22,26)(19,23,24,20)(21,25,29,32)(27,30,31,28);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,19,22,24)(18,20,26,23)(21,27,29,31)(25,28,32,30);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(21,29)(25,32)(27,31)(28,30);(2,6)(3,8)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839776);