C₂×#12207 #484241

order	512
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12207

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1019 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 146 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);(3,4,6,9)(5,7,10,8)(11,12)(13,21)(14,18)(15,19)(16,20)(17,23)(22,36)(24,37)(25,41)(26,33)(27,34)(28,42)(29,35)(30,38)(31,40)(32,39);(4,7)(5,10)(8,9)(11,13,16,24)(12,17,20,31)(14,25,27,22)(15,23,29,40)(18,32,34,30)(19,21,35,37)(26,39,28,38)(33,41,42,36);(11,14,16,27)(12,18,20,34)(13,22,24,25)(15,28,29,26)(17,30,31,32)(19,33,35,42)(21,36,37,41)(23,39,40,38)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391437);