C₂×#7047 #95330

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7047

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1966 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1966 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#7047⋊C₃
maximal subgroup of	#72268047, #72268048, #72268049, …
maximal normal subgroup of	#72268047, #72268048, #72268049, …
maximal quotient of	#72268048

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2526);