C₂×#13586 #512600

order	512
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₂×#13586

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1118 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1118 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)(5,14,16,9)(6,10)(7,12,22,11)(8,13)(15,30,31,23)(17,24,21,25)(18,32,33,26)(19,28,34,27)(20,29);(3,5,8,17)(4,9,13,25)(6,18,20,15)(7,21,22,16)(10,26,29,23)(11,24,12,14)(19,31,34,33)(27,30,28,32);(3,6,8,20)(4,10,13,29)(5,15,17,18)(7,19,22,34)(9,23,25,26)(11,27,12,28)(14,30,24,32)(16,31,21,33)

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, 419796);