#7656757

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

5452 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,28)(20,26)(21,22)(23,25)(24,27)(29,32)(30,31);(9,11,13,12)(10,14,15,16)(17,19,24,23)(18,25,27,28)(20,31,22,29)(21,30,26,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,20,24,22)(18,26,27,21)(19,29,23,31)(25,30,28,32);(17,21)(18,20)(19,30)(22,27)(23,32)(24,26)(25,31)(28,29);(1,4,5,3)(2,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,22,24,20)(18,21,27,26)(19,29,23,31)(25,30,28,32);(1,2)(3,6)(4,8)(5,7)(9,12,13,11)(10,14,15,16)(17,23,24,19)(18,25,27,28)(20,31,22,29)(21,32,26,30)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=gh,f^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, 7563953);