C₂²×#50635 #72324859

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

Aliases

direct product

C₂²×#50635

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

101177 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7768137, C₂²×#7737955
maximal normal subgroup of	C₂²×#7768137, C₂²×#7737955

GAP

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

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

C22×#50635 #72324859