C₂²×#59392 #72301124

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

Aliases

direct product

C₂²×#59392

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

82577 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)(17,18,20,22)(19,21,24,23);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,24)(22,23);(10,13)(11,15)(17,18,20,22)(19,21,24,23);(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24);(9,10)(11,16)(12,13)(14,15)(17,19)(18,21)(20,24)(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×#8063858, C₂²×#7687886, C₂²×#7677129
maximal normal subgroup of	C₂²×#8063858, C₂²×#7687886, C₂²×#7677129

GAP

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

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

C22×#59392 #72301124