C₂×#12500 #490733

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

Aliases

direct product

C₂×#12500

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

482 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 512

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

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

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, 397929);