#493681

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
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	C₂²×C₄⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1212 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)(5,6,10,16,11,15,27,34,12,17,28,29,9,26,19,35)(7,18,8,14,22,30,25,33,20,36,23,32,21,13,24,31);(5,7,9,21,12,20,11,22)(6,13,15,18,17,30,26,36)(8,19,24,28,23,27,25,10)(14,29,33,35,32,16,31,34);(2,4)(5,8)(6,14)(7,19)(9,24)(10,22)(11,25)(12,23)(13,29)(15,33)(16,30)(17,32)(18,35)(20,27)(21,28)(26,31)(34,36);(7,20)(8,23)(13,30)(14,32)(18,36)(21,22)(24,25)(31,33)

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,d^b=di,d^c=di

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, 400877);