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C₂×#21847 #6374403

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6490980, C₂×#6496144, C₂×#6374603, …
maximal normal subgroup of	C₂×#6490980, C₂×#6496144, C₂×#6374603, …
maximal quotient of	#72305088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281599);