#7679531

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

4851 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=ghi,e^2=gi,f^2=gh,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, 7586727);