#8496572

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

4966 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8403768);