C₂×#1099968 #72338020

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

Aliases

direct product

C₂×#1099968

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

34669 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232357, C₂×#72232291
maximal normal subgroup of	C₂×#72232281, C₂×#72232357, C₂×#72232291

GAP

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

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

C2×#1099968 #72338020