C₂×#7086 #122617

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7086

Subgroups

	subgroup	G/H
center, socle	C₂³	#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

688 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,20,17,21,16,32)(13,22,24,36,26,39,25,42)(14,27,29,19,31,37,30,38)(18,28,33,41,35,40,34,23);(4,7)(8,9)(11,13,17,26)(12,18,21,35)(14,28,31,40)(15,25,16,24)(19,36,38,42)(20,33,32,34)(22,37,39,27)(23,30,41,29)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29813);