#7726694

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

4761 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:

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

Polycyclic presentation:

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

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, 7633890);