C₂×#12164 #484630

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12164

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 995 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);(7,10)(11,12)(13,24)(14,18)(15,22)(16,21)(17,23)(19,26)(20,28)(25,38)(27,39)(29,36)(30,34)(31,35)(32,37)(33,42)(40,41);(3,4)(5,7)(6,8)(9,10)(12,18)(14,28)(15,16)(19,33)(20,22)(21,34)(23,35)(24,38)(25,41)(26,32)(29,42)(30,31)(36,39)(37,40);(3,5,6,9)(4,7,8,10)(11,13,17,27)(12,19,23,37)(14,29,31,25)(15,32,16,26)(18,33,35,40)(20,38,34,36)(21,39,22,24)(28,42,30,41)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391826);