C₂×#2189843 #72337985

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

Aliases

direct product

C₂×#2189843

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

36301 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)(8,11)(9,13)(15,16,19,17)(18,20,22,21)(23,24,27,25)(26,29,30,28);(3,4,6,5)(8,11)(12,14)(24,25)(26,30);(3,4)(5,6)(9,13)(12,14)(16,17)(18,22)(26,30)(28,29);(4,5)(7,8,10,11)(9,12,13,14)(15,17,19,16)(18,20,22,21)(23,25,27,24)(26,29,30,28);(1,2);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,18,19,22)(16,20,17,21)(23,26)(24,28)(25,29)(27,30)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232318, C₂×#72232336, C₂×#72232298
maximal normal subgroup of	C₂×#72232318, C₂×#72232336, C₂×#72232298

GAP

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

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

C2×#2189843 #72337985