C₂×#2839063 #72337967

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

Aliases

direct product

C₂×#2839063

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

37155 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)(7,8,10,11)(9,12,13,14)(16,19)(17,21)(23,24,27,29)(25,26,30,28);(3,4,6,5)(8,11)(9,13)(15,16,18,19)(17,22,21,20)(23,25,27,30)(24,28,29,26);(4,5)(17,21)(20,22)(25,30)(26,28);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,30);(1,2);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,17)(16,20)(18,21)(19,22)(23,26)(24,25)(27,28)(29,30)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232295, C₂×#72232308
maximal normal subgroup of	C₂×#72232281, C₂×#72232295, C₂×#72232308

GAP

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

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

C2×#2839063 #72337967