C₂×#719812 #72325554

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

Aliases

direct product

C₂×#719812

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

46405 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)(11,12,13,14)(15,16,19,17)(18,21,22,20);(8,9)(11,12,13,14)(15,16,19,17)(18,20,22,21);(16,17)(18,22);(4,6)(7,9,10,8)(11,12)(13,14)(15,17,19,16)(18,20,22,21);(1,2);(3,4)(5,6)(15,18,19,22)(16,20,17,21)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232156, C₂×#72232160, C₂×#72232185
maximal normal subgroup of	C₂×#72232156, C₂×#72232160, C₂×#72232185

GAP

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

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

C2×#719812 #72325554