#8093714

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

4642 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8000910);