C₂²×#41954 #72295880

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

Aliases

direct product

C₂²×#41954

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

78945 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);(10,13)(14,16)(18,22)(20,28)(23,25)(26,32);(6,8)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,29,28,24)(26,30,32,31);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,29)(21,22)(24,28)(25,27)(26,31)(30,32);(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(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), (10,13)(14,16)(18,22)(20,28)(23,25)(26,32), (6,8)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,29,28,24)(26,30,32,31), (9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,29)(21,22)(24,28)(25,27)(26,31)(30,32), (9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31), (5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32), (1,2)(3,4));

C22×#41954 #72295880