C₂×#1970557 #72337837

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

Aliases

direct product

C₂×#1970557

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

35343 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232318, C₂×#72232291, C₂×#72232342
maximal normal subgroup of	C₂×#72232318, C₂×#72232291, C₂×#72232342

GAP

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

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

C2×#1970557 #72337837