#8434223

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

5059 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341419);