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C₂×#8822 #191458

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	#384
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(4,7)(5,9)(8,10)(11,12,15,18,16,19,14,24)(13,20,22,25,23,26,21,17);(11,13,14,21,16,23,15,22)(12,17,18,20,19,25,24,26);(3,4,5,7,6,8,9,10)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225153, #72225163, C₂×#135758, …
maximal normal subgroup of	#72225153, #72225163, C₂×#135758, …
maximal quotient of	#72225156, #72225163, D₃×#8822, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98654);