C₂×#2078590 #72324995

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

Aliases

direct product

C₂×#2078590

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

50549 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232077, C₂×#72232094, C₂×#72232169, …
maximal normal subgroup of	C₂×#72232077, C₂×#72232094, C₂×#72232169, …

GAP

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

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

C2×#2078590 #72324995