#8588412

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

5144 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8495608);