#8458691

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

5050 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8365887);