C₂×#1337730 #72295014

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#1337730

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

37355 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293918
maximal normal subgroup of	C₂×#72293918
maximal quotient of	C₂×#72293928

GAP

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

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

C2×#1337730 #72295014