#8078336

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

5120 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)(9,10,13,16)(11,14,12,15)(17,18,23,20)(19,21,26,24)(22,25,30,28)(27,29,32,31);(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32);(1,2,4,6)(3,5,7,8)(19,26)(21,24)(27,32)(29,31);(3,7)(5,8)(9,11,13,12)(10,14,16,15)(17,20,23,18)(19,24,26,21)(22,25,30,28)(27,29,32,31);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,15,16,14)(17,21,23,24)(18,19,20,26)(22,29,30,31)(25,27,28,32);(1,2)(3,5)(4,6)(7,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,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, 7985532);