Finite Group Databasebeta

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C₂×#8302 #184740

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5526 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)(7,8,9,12,10,11,13,14)(15,16,18,21,19,22,17,20);(8,11)(9,13);(4,6)(15,17,19,18)(16,20,22,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292810, #72292814, #72309140, …
maximal normal subgroup of	#72292810, #72292814, #72309140, …
maximal quotient of	#72292827, #72293067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91936);