C₂×#918341 #72337886

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#918341

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

51989 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)(9,13)(10,11)(12,14);(3,4,6,5)(8,11)(9,14)(15,16)(17,20)(18,19)(21,22)(25,29)(27,30);(3,4)(5,6)(8,11)(9,14)(16,19)(20,22)(23,24,26,28)(25,30,29,27);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,30)(26,28)(27,29);(1,2);(3,5,6,4)(7,9)(8,12)(10,14)(11,13)(15,17,18,21)(16,20,19,22)(23,25)(24,27)(26,29)(28,30)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232360, C₂×#72232308, C₂×#72232330
maximal normal subgroup of	C₂×#72232360, C₂×#72232308, C₂×#72232330

GAP

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

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

C2×#918341 #72337886