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C₂×#11381 #358762

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

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

4946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#403941, #72261683, #72261705, …
maximal normal subgroup of	C₂×#403941, #72261683, #72261705, …
maximal quotient of	C₂×#149443, C₂×#122049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265958);