C₂×#12509 #491446

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₂×#12509

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 718 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) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,11)(6,14)(7,9)(8,12)(10,15)(13,17)(16,18)(20,27)(21,32)(22,23)(24,25)(28,33)(29,31)(30,34);(3,5,6,12,7,13,15,18)(4,8,10,11,9,16,14,17)(19,20,28,34,24,30,22,27,26,32,23,29,25,31,33,21);(4,9)(8,16)(10,14)(11,17)(19,21,33,31,25,29,23,32,26,27,22,30,24,34,28,20)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#12509, C₂×#491500, C₂×#123412, …
maximal normal subgroup of	C₂×#491500, C₂×#123412, C₂×#123419
maximal quotient of	D₃×#12509

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398642);