C₂×#582133 #72338017

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#582133

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

32285 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)(16,19)(17,21)(23,24)(25,27)(26,29)(28,30);(7,8)(9,14)(10,11)(12,13)(23,25,26,30)(24,27,29,28);(15,16)(17,20)(18,19)(21,22)(23,24)(25,28)(26,29)(27,30);(1,2);(7,8)(9,14)(10,11)(12,13)(15,16)(17,22)(18,19)(20,21)(23,24)(25,27)(26,29)(28,30);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,17,18,21)(16,20,19,22)(23,25)(24,28)(26,30)(27,29)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232279, C₂×#72232357, C₂×#72232292
maximal normal subgroup of	C₂×#72232279, C₂×#72232357, C₂×#72232292

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

C2×#582133 #72338017