#7643582

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

6506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550778);