#7916282

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

4624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7823478);