C₂²×#47800 #72295875

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

Aliases

direct product

C₂²×#47800

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

75289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 32 points:

(3,4);(9,10)(11,15)(12,14)(13,16)(19,25)(22,28)(26,31)(29,32);(6,8)(9,11)(10,15)(12,13)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32);(10,12)(13,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(9,12,14,10)(11,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(5,6)(7,8)(9,13,14,15)(10,16,12,11)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8037281
maximal normal subgroup of	C₂²×#8037281

GAP

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

gap> G := Group((3,4), (9,10)(11,15)(12,14)(13,16)(19,25)(22,28)(26,31)(29,32), (6,8)(9,11)(10,15)(12,13)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32), (10,12)(13,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32), (9,12,14,10)(11,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32), (5,6)(7,8)(9,13,14,15)(10,16,12,11)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32), (1,2)(3,4));

C22×#47800 #72295875