C₂×#4607160 #72337932

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#4607160

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

44373 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232308, C₂×#72232342
maximal normal subgroup of	C₂×#72232308, C₂×#72232342

GAP

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

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

C2×#4607160 #72337932