C₂×#11994 #483308

order	512
nilpotency class	4
isoclinism	isoclinic class of #11994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#11994

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

662 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 64

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#483311, C₂×#133271, C₂×#133275
maximal normal subgroup of	C₂×#483311, C₂×#133271, C₂×#133275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390504);