C₂×#15467 #609040

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

direct product

C₂×#15467

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

11994 conjugacy classes of subgroups; only the 1010 normal subgroups are shown.

normal subgroups only

This group has 11994 conjugacy classes of subgroups — too many to draw. Only its 1010 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) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(4,6)(7,8)(9,10)(11,12,13,14)(15,16,18,20)(17,19,21,22);(7,8,9,10)(11,12,13,14)(17,21)(19,22);(7,8)(9,10)(11,12)(13,14)(15,17)(16,19)(18,21)(20,22);(3,4)(5,6)(7,8)(9,10)(15,16)(17,19)(18,20)(21,22)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#15467⋊C₃
maximal subgroup of	#72210048, #72219199, #72267005, …
maximal normal subgroup of	#72210048, #72219199, #72267005, …
maximal quotient of	C₂×#432839, C₂×#432838, C₂×#432837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516236);