C₂×#1517202 #72337845

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#1517202

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

38029 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232279, C₂×#72232336, C₂×#72232314
maximal normal subgroup of	C₂×#72232279, C₂×#72232336, C₂×#72232314

GAP

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

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

C2×#1517202 #72337845