#8063114

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

4871 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,30)(28,29)(31,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,16)(10,14,12,15)(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30);(3,7)(4,8)(9,12,13,10)(11,15,16,14)(18,25)(22,32)(24,26)(28,30);(2,6)(4,8)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28);(1,4)(2,3)(5,8)(6,7)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30);(1,2)(3,4)(5,6)(7,8)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7970310);