C₂×#13429 #499116

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13429

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

1161 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1161 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)(5,14)(6,9)(7,12)(8,13)(10,17)(11,18)(15,27)(16,31)(19,23)(20,25)(21,28)(22,32)(24,34)(26,33)(29,30);(4,9)(5,15)(6,18)(10,22)(11,13)(12,23)(14,29)(19,33)(24,25)(26,28)(27,31)(32,34);(3,5,6,16,8,15,18,30)(4,10,11,24,13,25,9,22)(7,20,19,34,21,17,33,32)(12,27,26,31,28,14,23,29)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406312);