#8475684

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

4944 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8382880);