#494231

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

990 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:

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

Polycyclic presentation:

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

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, 401427);