#8528908

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

5049 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)(11,12)(14,15)(17,18,23,25)(19,24,21,29)(20,26,30,31)(22,27,32,28);(2,6)(3,8)(9,10,13,16)(11,14,12,15)(17,19,23,21)(18,24,25,29)(20,27,30,28)(22,26,32,31);(1,3)(2,4)(5,8)(6,7)(9,11,13,12)(10,14,16,15)(18,25)(19,21)(26,31)(27,28);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,26)(19,28)(21,27)(22,24)(23,30)(25,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,12,13,11)(10,15,16,14)(17,21,23,19)(18,24,25,29)(20,27,30,28)(22,31,32,26);(17,22)(18,27)(19,26)(20,29)(21,31)(23,32)(24,30)(25,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8436104);