C₂×#4311750 #72338015

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

Aliases

direct product

C₂×#4311750

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

36029 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)(15,16,18,20)(17,21,22,19)(23,24,27,26)(25,29,30,28);(3,4,6,5)(7,8,10,11)(9,14,13,12)(15,16,18,20)(17,19,22,21)(23,25,27,30)(24,28,26,29);(4,5)(9,13)(12,14)(17,22)(19,21)(24,26)(28,29);(1,2);(7,8)(9,14)(10,11)(12,13)(15,16)(17,21)(18,20)(19,22)(23,25)(24,28)(26,29)(27,30);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,17)(16,19)(18,22)(20,21)(23,26,27,24)(25,29,30,28)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232318, C₂×#72232287
maximal normal subgroup of	C₂×#72232281, C₂×#72232318, C₂×#72232287

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

C2×#4311750 #72338015