C₂×#625786 #72299231

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

Aliases

direct product

C₂×#625786

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

69377 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298074, C₂×#72213999, C₂×#72298091
maximal normal subgroup of	C₂×#72298074, C₂×#72213999, C₂×#72298091

GAP

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

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

C2×#625786 #72299231