#7656974

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

4830 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564170);