C₂×#13436 #499122

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#13436

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	C₂×#3199	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1169 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1169 conjugacy classes of subgroups — too many to draw. Only its 178 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)(5,15)(6,10)(7,21)(8,12)(9,14)(11,18)(13,22)(16,29)(17,30)(19,31)(20,28)(23,25)(24,33)(26,32)(27,34);(4,10)(5,16)(6,8)(7,22)(12,14)(13,28)(17,19)(18,32)(21,25)(24,26)(29,30)(33,34);(3,5,6,17,9,19,8,16)(4,11,12,26,14,27,10,24)(7,18,20,33,22,34,23,32)(13,15,28,29,21,31,25,30)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#500810, #72311877, #72311884, …
maximal normal subgroup of	C₂×#500810, #72311877, #72311884, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406318);