C₂×#2077566 #72337930

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

Aliases

direct product

C₂×#2077566

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

36301 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)(7,8,10,12)(9,11,13,14)(15,16,18,20)(17,19,21,22)(23,24,26,29)(25,28,30,27);(15,17,18,21)(16,19,20,22)(23,25)(24,27)(26,30)(28,29);(3,4)(5,6)(9,13)(11,14)(23,24,26,29)(25,27,30,28);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,29)(28,30);(1,2);(7,9,10,13)(8,11,12,14)(15,17)(16,19)(18,21)(20,22)(23,25)(24,28)(26,30)(27,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232295, C₂×#72232314, C₂×#72232342
maximal normal subgroup of	C₂×#72232295, C₂×#72232314, 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)(7,8,10,12)(9,11,13,14)(15,16,18,20)(17,19,21,22)(23,24,26,29)(25,28,30,27), (15,17,18,21)(16,19,20,22)(23,25)(24,27)(26,30)(28,29), (3,4)(5,6)(9,13)(11,14)(23,24,26,29)(25,27,30,28), (3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,29)(28,30), (1,2), (7,9,10,13)(8,11,12,14)(15,17)(16,19)(18,21)(20,22)(23,25)(24,28)(26,30)(27,29));

C2×#2077566 #72337930