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C₂×#8218 #182595

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#11711005, #72217661, #72217673, …
maximal normal subgroup of	#72217661, #72217673, C₂×#208392, …
maximal quotient of	C₂×#96738, C₂×#128834, C₂×#128835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89791);