C₂×#4861908 #72296399

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#4861908

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

33269 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231820
maximal normal subgroup of	C₂×#72231820
maximal quotient of	C₂×#72278194

GAP

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

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

C2×#4861908 #72296399