#7645326

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

5143 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)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,28)(31,32);(1,2,5,6)(3,7,4,8)(10,13)(14,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30);(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32);(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32);(1,4,5,3)(2,8,6,7)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,f^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, 7552522);