C₂²×#49984 #72294286

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

Aliases

direct product

C₂²×#49984

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

71125 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293931
maximal normal subgroup of	C₂×#72293931
maximal quotient of	C₂²×#4940920

GAP

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

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

C22×#49984 #72294286