C₂²×#59038 #72291844

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

Aliases

direct product

C₂²×#59038

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

99017 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)(10,13)(11,15)(18,19)(22,23);(17,18,21,19)(20,23,24,22);(10,13)(14,16)(18,19)(22,23);(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,22,24,23);(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23);(3,4);(1,2)(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7837443, C₂²×#7850713, C₂²×#7699591
maximal normal subgroup of	C₂²×#7837443, C₂²×#7850713, C₂²×#7699591
maximal quotient of	C₂²×#647855

GAP

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

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

C22×#59038 #72291844