#8091794

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

5360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7998990);