#8078331

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

5079 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)(3,5)(4,6)(7,8)(10,14)(15,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,30)(29,31);(1,2,4,6)(3,5,7,8)(9,11,13,12)(10,15,14,16)(18,23)(19,26)(24,29)(27,28);(3,7)(5,8)(17,20)(18,24)(19,27)(21,30)(22,31)(23,29)(25,32)(26,28);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,16,14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,31)(28,30)(29,32);(1,2)(3,5)(4,6)(7,8)(17,18)(19,22)(20,29)(21,23)(24,30)(25,26)(27,32)(28,31)

Polycyclic presentation:

2^9|c^2=gh,e^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=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, 7985527);