#7645316

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

5006 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7552512);