#7683428

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

5400 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7590624);