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C₂×#10761 #353329

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂¹⁰

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

6754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219233, #72225621, #72225888, …
maximal normal subgroup of	#72219233, #72225621, #72225888, …
maximal quotient of	C₂×#106076, C₂×#146667, C₂×#106654, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260525);