C₂×#7305 #127999

order	512
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#7305

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1440 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 154 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)(6,8)(7,10)(11,12)(13,21)(14,25,26,18)(15,19)(16,20)(17,22,29,35)(23,33)(24,34)(27,38,39,31)(28,32)(30,36,40,42)(37,41);(3,5)(4,7,8,9)(6,10)(11,13,15,23,16,24,28,37)(12,17,19,30,20,29,32,40)(14,22,27,36,26,35,39,42)(18,21,31,33,25,34,38,41)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268051, #72268052
maximal normal subgroup of	#72268051, #72268052

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35195);