C₂×#2620887 #72337856

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#2620887

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

35519 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232336, C₂×#72232330
maximal normal subgroup of	C₂×#72232281, C₂×#72232336, C₂×#72232330

GAP

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

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

C2×#2620887 #72337856