C₂×#2822359 #72294597

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

Aliases

direct product

C₂×#2822359

Subgroups

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

43173 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)(9,10)(11,12)(13,14)(15,16,19,21)(17,22,18,20)(23,24,27,26)(25,28,30,29);(7,9,11,13)(8,10,12,14)(15,17,19,18)(16,20,21,22)(23,25)(24,28)(26,29)(27,30);(4,6)(8,12)(9,13)(16,21)(20,22)(23,25,27,30)(24,29,26,28);(7,9)(8,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,21)(23,25)(24,29)(26,28)(27,30);(3,4)(5,6)(7,10)(8,9)(11,14)(12,13)(15,18,19,17)(16,20,21,22)(23,26,27,24)(25,29,30,28);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293940, C₂×#72293948, C₂×#72293922, …
maximal normal subgroup of	C₂×#72293940, C₂×#72293948, C₂×#72293922, …
maximal quotient of	C₂×#72293934

GAP

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

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

C2×#2822359 #72294597