C₂×#2806689 #72294562

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

Aliases

direct product

C₂×#2806689

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

35345 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912, C₂×#72230859
maximal normal subgroup of	C₂×#72293912, C₂×#72230859
maximal quotient of	C₂×#72278753

GAP

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

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

C2×#2806689 #72294562