C₂²×#59020 #72299816

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

Aliases

direct product

C₂²×#59020

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

94633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7696044
maximal normal subgroup of	C₂²×#7696044

GAP

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

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

C22×#59020 #72299816