C₂×#11917 #483019

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#11917

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 194 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);(3,4,7,11,6,12,19,30,8,13,23,31,20,10,22,28)(5,14,16,32,17,9,24,26,18,25,33,29,15,27,21,34);(3,5)(4,9)(6,15)(7,21)(8,18)(10,25)(11,29)(12,14)(13,27)(16,22)(17,20)(19,33)(23,24)(26,30)(28,34)(31,32);(3,6,8,20)(4,10,13,12)(5,15,18,17)(7,22,23,19)(9,25,27,14)(11,30,31,28)(16,24,33,21)(26,32,34,29)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#483023, C₂×#133271, C₂×#133265
maximal normal subgroup of	C₂×#483023, C₂×#133271, C₂×#133265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390215);