C₂×#616963 #72342020

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#616963

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

75085 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72310669
maximal normal subgroup of	C₂×#72310669

GAP

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

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

C2×#616963 #72342020