C₂²×#35278 #72301144

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

Aliases

direct product

C₂²×#35278

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

93517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7779660, C₂²×#7779984, C₂²×#7671954
maximal normal subgroup of	C₂²×#7779660, C₂²×#7779984, C₂²×#7671954

GAP

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

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

C22×#35278 #72301144