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C₂×#9495 #201648

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

4070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#245225, #72218846, #72219460, …
maximal normal subgroup of	C₂×#245225, #72218846, #72219460, …
maximal quotient of	C₂×#139731, C₂×#139743, C₂×#139750, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108844);