C₂×#12522 #491476

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#12522

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
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

538 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 218 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)(5,11,14,18)(6,15,12,9)(8,16,17,13)(19,20)(21,29)(22,25)(23,36)(24,37)(26,32)(27,30)(28,35)(31,41)(33,48)(34,49)(38,46)(39,45)(40,50)(42,44)(43,47);(3,5,12,16,7,14,6,13)(4,8,15,18,10,17,9,11)(20,25)(26,38)(27,39)(28,37)(29,41)(36,49)(40,42)(43,48);(19,21,30,44,24,33,34,46,22,31,45,50,35,47,23,32)(20,26,36,43,28,40,39,41,25,38,49,48,37,42,27,29)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398672);