Finite Group Databasebeta

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C₄×#1313 #126523

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4);(13,14,15,19,16,21,23,27,17,20,22,28,24,18,25,26);(5,6,7,9,8,10,11,12)(14,18,20,21)(15,22)(19,27,28,26)(23,25)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216161, #72216172, #72216174, …
maximal normal subgroup of	#72216161, #72216172, #72216174, …
maximal quotient of	#72216161, #72216178, C₈×#1313, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33719);