C₂×#4308429 #72299428

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#4308429

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

59125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297599, C₂×#72297595
maximal normal subgroup of	C₂×#72297599, C₂×#72297595

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

C2×#4308429 #72299428