C₂×#12502 #490734

order	512
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12502

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 50
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4,7,10)(5,11,13,18)(6,15,12,9)(8,16,17,14)(19,20)(21,29)(22,25)(23,35)(24,38)(26,31)(27,37)(28,34)(30,45)(32,49)(33,46)(36,44)(39,43)(40,47)(41,48)(42,50);(3,5,12,16,7,13,6,14)(4,8,15,18,10,17,9,11)(20,25)(26,40)(27,41)(28,38)(29,45)(35,46)(42,44)(43,49);(5,13)(8,17)(11,18)(14,16)(19,21,22,30)(20,26,25,40)(23,36,33,50)(24,39,34,32)(27,43,41,49)(28,44,38,42)(29,46,45,35)(31,48,47,37)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397930);