C₂²×#59375 #72301115

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

Aliases

direct product

C₂²×#59375

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

90857 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7763450, C₂²×#8062966, C₂²×#7699591
maximal normal subgroup of	C₂²×#7763450, C₂²×#8062966, C₂²×#7699591

GAP

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

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

C22×#59375 #72301115