#348517

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,13)(5,10)(6,15)(7,11)(9,14)(12,16)(17,18,23,27)(19,28,26,22)(20,29,34,41)(21,35,39,25)(24,32,42,33)(30,37,36,38)(31,43,46,44)(40,45,48,47);(1,3,7,4)(2,8,11,13)(5,14,6,12)(9,15,16,10)(17,19,22,18,23,26,28,27)(20,29,32,24,34,41,33,42)(21,36,37,35,39,30,38,25)(31,44,45,48,46,43,47,40);(1,4,7,3)(2,9,11,16)(5,14,6,12)(8,15,13,10)(17,20,23,34)(18,24,27,42)(19,29,26,41)(21,31,39,46)(22,32,28,33)(25,40,35,48)(30,43,36,44)(37,45,38,47);(1,5,7,6)(2,10,11,15)(3,12,4,14)(8,16,13,9)(17,21,23,39)(18,25,27,35)(19,30,26,36)(20,31,34,46)(22,37,28,38)(24,40,42,48)(29,43,41,44)(32,45,33,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255713);