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C₂×#8813 #191416

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#8813⋊C₃, C₂×#212918, #72214381, …
maximal normal subgroup of	C₂×#8813⋊C₃, C₂×#212918, #72214381, …
maximal quotient of	C₂×#121387, #72224817, #72225153, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98612);