#7682539

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

4733 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,f^2=g,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, 7589735);