C₂×#581540 #72296404

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#581540

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

31785 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231820
maximal normal subgroup of	C₂×#72231820
maximal quotient of	C₂×#72278103

GAP

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

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

C2×#581540 #72296404