#8271125

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

5609 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8178321);