#7744748

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

4956 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:

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

Polycyclic presentation:

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

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, 7651944);