#9041001

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

4952 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)(3,5)(4,8)(6,7)(10,12)(14,15)(17,18,22,24)(19,23,26,20)(21,29,30,25)(27,28,32,31);(2,7)(3,8)(9,10,13,12)(11,15,16,14)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32);(1,3,6,8)(2,4,7,5)(9,11,13,16)(10,14,12,15)(18,24)(19,26)(21,30)(28,31);(1,4,6,5)(2,3,7,8)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,2)(3,4)(5,8)(6,7)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32);(1,5,6,4)(2,8,7,3)(17,21,22,30)(18,25,24,29)(19,27,26,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948197);