C₂×#988863 #72299321

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

Aliases

direct product

C₂×#988863

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

35403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298096, C₂×#72298094, C₂×#72298081
maximal normal subgroup of	C₂×#72298096, C₂×#72298094, C₂×#72298081

GAP

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

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

C2×#988863 #72299321