#8178685

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

5011 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8085881);