#359174

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 298 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,18,14,17,24,30)(11,15,21,27,20,25,33,36)(12,16,22,29,23,28,34,38)(19,26,32,37,31,35,39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,25)(18,27)(22,32)(23,31)(24,33)(28,35)(29,37)(30,36)(34,39)(38,40);(15,25)(26,35)(27,36)(37,40);(1,4,5,3)(2,6,7,8)(9,12,13,22,14,23,24,34)(10,16,18,29,17,28,30,38)(11,19,21,32,20,31,33,39)(15,26,27,37,25,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266370);