#8508948

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

5175 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)(11,15)(13,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(19,20)(21,31)(24,28)(25,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,27,28)(21,26,31,29)(22,25,32,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,24,27,28)(21,29,31,26)(22,25,32,30);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,31)(18,25,27,30)(19,26,20,29)(22,24,32,28);(17,22,23,32)(18,26,27,29)(19,25,20,30)(21,24,31,28)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=gi,e^2=i,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, 8416144);