C₂×#17339 #6342468

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#17339

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₄²	C₂⁵
socle	C₂³	C₂²×C₄²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2068 conjugacy classes of subgroups; only the 1538 normal subgroups are shown.

normal subgroups only

This group has 2068 conjugacy classes of subgroups — too many to draw. Only its 1538 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) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249664);