C₂×#582851 #72294648

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

Aliases

direct product

C₂×#582851

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

44237 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232697
maximal normal subgroup of	C₂×#72232697
maximal quotient of	C₂×#72293972

GAP

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

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

C2×#582851 #72294648