#8012213

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

4990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,e^2=i,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7919409);