#7651240

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

5127 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)(10,13)(11,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,30)(31,32);(1,2,4,5)(3,6,7,8)(10,13)(14,16)(17,19)(18,23)(20,21)(22,28)(24,32)(25,29)(26,27)(30,31);(9,10,12,13)(11,14,15,16)(19,28)(21,31)(23,27)(25,32);(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,21)(22,30)(23,32)(25,27)(26,29)(28,31);(1,3,4,7)(2,6,5,8)(17,18)(19,23)(20,29)(21,25)(22,26)(24,30)(27,28)(31,32);(17,21)(18,25)(19,20)(22,31)(23,29)(24,27)(26,32)(28,30)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558436);