C₂×#993567 #72294416

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

Aliases

direct product

C₂×#993567

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

47531 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293910
maximal normal subgroup of	C₂×#72293910
maximal quotient of	C₂×#72293945

GAP

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

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

C2×#993567 #72294416