C₂×#7058 #122258

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7058

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂³	#323
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

1108 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 98 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)(11,12,15,20,16,21,30,38,17,22,31,39,14,19,28,36)(13,23,25,40,24,34,41,37,26,35,29,42,27,18,32,33);(4,7)(8,9)(11,13)(12,18)(14,27)(15,29)(16,24)(17,26)(19,35)(20,37)(21,23)(22,34)(25,31)(28,41)(30,32)(33,39)(36,40)(38,42)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29454);