C₂×#11504 #480248

order	512
nilpotency class	4
isoclinism	isoclinic class of #11502
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#11504

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1570 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,21)(14,18)(15,28,29,19)(16,20)(17,23,31,38)(22,35)(24,36)(25,39)(26,41,42,33)(27,34)(30,37,32,40);(11,13,16,24)(12,17,20,31)(14,25,27,22)(15,23,29,38)(18,32,34,30)(19,21,28,36)(26,40,42,37)(33,39,41,35);(4,7)(8,10)(11,14,16,27)(12,18,20,34)(13,22,24,25)(15,26,29,42)(17,30,31,32)(19,33,28,41)(21,35,36,39)(23,37,38,40)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387444);