C₂×#7650 #160948

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7650

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

978 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 282 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) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68144);