#493691

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,10,17,11,15,21,31,12,16,20,13,9,27,7,18)(8,22,25,32,26,36,19,34,23,14,28,29,24,30,35,33);(1,2,3,4)(5,7,9,20,12,21,11,10)(6,13,15,18,16,17,27,31)(8,19,24,25,23,35,26,28)(14,29,30,33,22,32,36,34);(5,8)(6,14)(7,19)(9,24)(10,28)(11,26)(12,23)(13,29)(15,30)(16,22)(17,32)(18,33)(20,25)(21,35)(27,36)(31,34);(8,23)(14,22)(19,35)(24,26)(25,28)(29,32)(30,36)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^c=di,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400887);