#8844502

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

5191 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:

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

Polycyclic presentation:

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

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, 8751698);