#300410

order	512
nilpotency class	3
isoclinism	isoclinic class of #300397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3559	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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)(5,14)(6,9)(7,10)(11,15)(12,16)(17,18)(19,28)(20,33)(21,35)(22,38)(23,27)(24,31)(25,36)(26,39)(29,30)(32,43)(34,47)(37,44)(40,45)(41,48)(42,46);(1,3,7,12)(2,8,10,16)(4,5,11,6)(9,13,14,15)(17,19,22,31,23,32,39,48)(18,24,26,28,27,41,38,43)(20,34,35,42,36,29,44,45)(21,30,33,46,37,47,25,40);(1,4,5,3,7,11,6,12)(2,8,9,13,10,16,14,15)(17,20)(18,25)(19,29,32,34)(21,26)(22,35)(23,36)(24,40,41,46)(27,33)(28,30,43,47)(31,45,48,42)(37,38)(39,44);(17,21,22,33,23,37,39,25)(18,20,26,35,27,36,38,44)(19,30,31,46,32,47,48,40)(24,34,28,42,41,29,43,45)

Polycyclic presentation:

2^9|b^2=ei,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,g^c=ghi

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, 207606);