C₂×#4320270 #72338032

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#4320270

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

35633 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)(7,8,10,12)(9,13,14,11)(16,19)(17,21)(23,24)(25,27)(26,29)(28,30);(15,16)(17,20)(18,19)(21,22)(23,25,26,30)(24,27,29,28);(3,4)(5,6)(9,14)(11,13)(17,21)(20,22)(25,30)(27,28);(1,2);(7,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,29)(28,30);(7,9,10,14)(8,11,12,13)(15,17)(16,20)(18,21)(19,22)(23,25)(24,28)(26,30)(27,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232342, C₂×#72232287, C₂×#72232286
maximal normal subgroup of	C₂×#72232342, C₂×#72232287, C₂×#72232286

GAP

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

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

C2×#4320270 #72338032