#8497879

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

5096 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,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, 8405075);