C₂×#1419405 #72294584

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#1419405

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35493 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)(7,8,10,12)(9,13,14,11)(15,16,18,20)(17,21,22,19)(23,24)(25,29)(26,28)(27,30);(9,14)(11,13)(15,16)(17,19)(18,20)(21,22)(23,24,26,28)(25,27,30,29);(15,16,18,20)(17,19,22,21)(23,25)(24,27)(26,30)(28,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,25)(24,27)(26,30)(28,29);(7,9)(8,11)(10,14)(12,13)(15,17)(16,19)(18,22)(20,21)(23,24)(25,27)(26,28)(29,30);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912
maximal normal subgroup of	C₂×#72293912
maximal quotient of	C₂×#72293987

GAP

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

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

C2×#1419405 #72294584