C₂×#7918533 #72304385

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7918533

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 9798 conjugacy classes of subgroups; only the 31853 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72304329
maximal normal subgroup of	C₂×#72304329

GAP

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

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

C2×#7918533 #72304385