#7656796

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

5041 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)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,26,28,22)(20,24,31,32)(21,29,30,25);(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,29)(21,24)(23,28)(25,31)(26,27)(30,32);(1,3,5,4)(2,6,7,8)(17,20,23,31)(18,24,27,32)(19,25,28,29)(21,22,30,26);(9,11,12,15)(10,13,14,16)(17,21,23,30)(18,25,27,29)(19,24,28,32)(20,22,31,26);(1,4,5,3)(2,6,7,8)(17,22)(18,19)(20,30)(21,31)(23,26)(24,25)(27,28)(29,32);(1,2)(3,6)(4,8)(5,7)(17,19)(18,26)(20,25)(21,24)(22,27)(23,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=g,e^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, 7563992);