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C₂×#7846 #180810

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂²	C₂×C₄³
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1466 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);(7,8,9,12,10,11,13,14);(15,16,17,20,18,19,21,22);(3,4,5,6)(8,11)(12,14)(16,19)(20,22)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#7846⋊C₃, C₂×#212918, C₂×#213216, …
maximal normal subgroup of	C₂×#7846⋊C₃, C₂×#212918, C₂×#213216, …
maximal quotient of	C₂×#105250, #72224814, C₂×#126544, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88006);