#7681858

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

5049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=i,e^2=gh,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589054);