C₂²×#46829 #72295884

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

Aliases

direct product

C₂²×#46829

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

77337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

C22×#46829 #72295884