#8471189

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

5089 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)(4,5)(6,8)(10,13)(11,15)(17,18,23,20)(19,24,30,27)(21,31,29,25)(22,26,28,32);(1,3)(2,7)(4,8)(5,6)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,27,30,24)(21,25,29,31)(22,26,28,32);(1,5,6,3)(2,4,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,27,30,24)(21,31,29,25)(22,26,28,32);(9,11)(10,14)(12,15)(13,16)(17,21,23,29)(18,25,20,31)(19,28,30,22)(24,26,27,32);(1,4)(2,5)(3,8)(6,7)(17,22,23,28)(18,26,20,32)(19,29,30,21)(24,25,27,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=gi,e^2=hi,f^2=hi,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, 8378385);