C₂×#874584 #72325152

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#874584

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232088, C₂×#72232188, C₂×#72232102, …
maximal normal subgroup of	C₂×#72232088, C₂×#72232188, C₂×#72232102, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×#874584 #72325152