#8097733

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

5663 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004929);