C₂×#1929597 #72338024

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#1929597

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

35209 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232283, C₂×#72232291, C₂×#72232286
maximal normal subgroup of	C₂×#72232283, C₂×#72232291, C₂×#72232286

GAP

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

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

C2×#1929597 #72338024