C₂²×#49849 #72296201

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

Aliases

direct product

C₂²×#49849

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

83265 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)(10,11)(13,14)(15,19)(16,17)(18,20)(23,27)(25,28);(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,28,27,25);(14,17)(18,19)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9228811, C₂²×#8387575
maximal normal subgroup of	C₂²×#9228811, C₂²×#8387575
maximal quotient of	C₂²×#1268761, C₂²×#1083597

GAP

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

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

C22×#49849 #72296201