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C₂×#1513 #17555

order	256
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#106058, #107850, #107878, …
maximal normal subgroup of	#106058, #107850, #107878, …
maximal quotient of	C₂×#7597, C₂×#7598, C₂×#7599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10543);