C₂²×#59032 #72324611

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

Aliases

direct product

C₂²×#59032

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

95157 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)(11,16)(12,14)(13,15)(18,21)(19,23);(9,11,13,14)(10,12,15,16)(17,18,20,21)(19,24,23,22);(6,8)(9,10,13,15)(11,16,14,12)(17,18)(19,22)(20,21)(23,24);(3,4);(6,8)(9,12,13,16)(10,14,15,11)(17,19,20,23)(18,22,21,24);(1,2)(3,4);(5,6)(7,8)(9,10)(11,16)(12,14)(13,15)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7677211
maximal normal subgroup of	C₂²×#7677211

GAP

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

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

C22×#59032 #72324611