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C₂×#13080 #494672

order	512
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233604, #72233629, #72233641, …
maximal normal subgroup of	#72233604, #72233629, #72233641, …
maximal quotient of	#72233597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401868);