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C₂×#12727 #493461

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#511853, #72237966, #72238631, …
maximal normal subgroup of	C₂×#511853, #72237966, #72238631, …
maximal quotient of	#72237881, #72249504, #72249667, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400657);