Finite Group Databasebeta

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C₂×#8328 #185001

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#121	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5966 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)(7,8)(9,13,14,11)(10,12)(15,16)(17,19)(18,21)(20,22);(7,9)(8,11)(10,14)(12,13)(16,19)(17,22)(20,21);(4,6)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238063, #72260952, #72260955, …
maximal normal subgroup of	#72238063, #72260952, #72260955, …
maximal quotient of	C₂×#129729, #72261493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92197);