C₂²×#58260 #72326138

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#58260

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

90149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8387575
maximal normal subgroup of	C₂²×#8387575
maximal quotient of	C₂²×#2805677

GAP

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

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

C22×#58260 #72326138