C₂×#12201 #484798

order	512
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12201

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

995 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 995 conjugacy classes of subgroups — too many to draw. Only its 138 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);(4,8)(5,6)(9,10)(11,12)(13,17)(14,24)(15,28)(16,20)(18,27)(19,29)(21,36)(22,35)(23,32)(25,40)(26,39)(30,37)(31,38)(33,41)(34,42);(3,4)(5,8)(6,10)(7,9)(11,13,16,23)(12,17,20,32)(14,25,26,21)(15,22,29,38)(18,33,34,30)(19,35,28,31)(24,36,39,40)(27,41,42,37);(3,5,7,6)(4,8,9,10)(11,14)(12,18,20,34)(13,21)(15,27)(16,26)(17,30,32,33)(19,24,28,39)(22,37)(23,25)(29,42)(31,36,35,40)(38,41)

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, 391994);