C₂×#2171340 #72296392

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

Aliases

direct product

C₂×#2171340

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

33893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(9,13)(12,14)(16,20)(18,21)(24,28)(26,29);(3,4,6,5)(7,8,10,11)(9,12,13,14)(15,16)(17,21)(18,22)(19,20)(23,24,27,28)(25,29,30,26);(3,4,6,5)(8,11)(9,13)(16,20)(18,21)(23,25,27,30)(24,29,28,26);(4,5)(7,8)(9,14)(10,11)(12,13)(15,17)(16,21)(18,20)(19,22)(23,25)(24,26)(27,30)(28,29);(3,5,6,4)(7,9,10,13)(8,12,11,14)(15,18,19,21)(16,22,20,17)(23,26)(24,30)(25,28)(27,29);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232907
maximal normal subgroup of	C₂×#72232907
maximal quotient of	C₂×#72278160

GAP

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

gap> G := Group((9,13)(12,14)(16,20)(18,21)(24,28)(26,29), (3,4,6,5)(7,8,10,11)(9,12,13,14)(15,16)(17,21)(18,22)(19,20)(23,24,27,28)(25,29,30,26), (3,4,6,5)(8,11)(9,13)(16,20)(18,21)(23,25,27,30)(24,29,28,26), (4,5)(7,8)(9,14)(10,11)(12,13)(15,17)(16,21)(18,20)(19,22)(23,25)(24,26)(27,30)(28,29), (3,5,6,4)(7,9,10,13)(8,12,11,14)(15,18,19,21)(16,22,20,17)(23,26)(24,30)(25,28)(27,29), (1,2));

C2×#2171340 #72296392