C₂×#2838959 #72337893

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

Aliases

direct product

C₂×#2838959

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

36883 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)(9,11)(10,12)(13,14)(15,16,19,20)(17,21,22,18)(23,24)(25,29)(26,28)(27,30);(3,4,6,5)(8,9)(12,13)(17,22)(18,21)(23,25,28,30)(24,29,26,27);(4,5)(8,9)(12,13)(24,26)(27,29);(4,5)(7,9,11,8)(10,12,14,13)(15,17)(16,18)(19,22)(20,21)(23,26,28,24)(25,29,30,27);(1,2);(3,5,6,4)(7,10)(8,12)(9,13)(11,14)(15,18)(16,17)(19,21)(20,22)(23,27,28,29)(24,25,26,30)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232318, C₂×#72232334, C₂×#72232297
maximal normal subgroup of	C₂×#72232318, C₂×#72232334, C₂×#72232297

GAP

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

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

C2×#2838959 #72337893