#8760994

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

4908 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8668190);