#7642474

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

5066 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549670);