C₂×#720093 #72325138

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

Aliases

direct product

C₂×#720093

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

54029 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232094, C₂×#72232147, C₂×#72232131
maximal normal subgroup of	C₂×#72232094, C₂×#72232147, C₂×#72232131

GAP

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

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

C2×#720093 #72325138