Finite Group Databasebeta

← Prev (#195542) Next (#195544) →

C₂×#9061 #195543

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#209860, #72226850, C₂×#210060, …
maximal normal subgroup of	C₂×#209860, #72226850, C₂×#210060, …
maximal quotient of	#72224802, #72224809, #72226786, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102739);