C₂²×#35446 #72339331

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#35446

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

312421 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7630442
maximal normal subgroup of	C₂²×#7630442
maximal quotient of	C₂×#72337225, C₂²×#516323

GAP

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

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

C22×#35446 #72339331