#7656815

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

5056 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=gh,f^2=h,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, 7564011);