#8905375

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

5111 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8812571);