C₂×#616962 #72341990

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

Aliases

direct product

C₂×#616962

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

77541 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)(18,25)(21,23)(26,30)(28,29);(1,2)(7,8)(9,10)(12,14)(15,16,20,17)(18,21,25,23)(19,22,27,24)(26,28,30,29);(7,8)(9,10)(12,14)(15,17,20,16)(18,23,25,21)(19,24,27,22)(26,29,30,28);(11,12)(13,14)(15,17)(16,20)(18,21)(19,22)(23,25)(24,27)(26,29)(28,30);(3,4)(5,6)(7,8)(9,10)(15,18)(16,21)(17,23)(19,26)(20,25)(22,28)(24,29)(27,30);(3,4)(5,6)(7,8)(9,10)(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)

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

C2×#616962 #72341990