#8756093

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

4913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663289);