C₂²×#48487 #72296207

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#48487

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

79245 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9228811
maximal normal subgroup of	C₂²×#9228811
maximal quotient of	C₂²×#2310627

GAP

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

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

C22×#48487 #72296207