#299289

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
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	#18646	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

911 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,30)(20,25)(21,36)(22,28)(23,29)(24,32)(26,37)(27,34)(31,45)(33,40)(35,39)(38,42)(41,47)(43,48)(44,46);(17,19,22,32,23,20,34,33)(18,24,27,30,29,40,28,25)(21,31,37,46,38,35,48,47)(26,39,36,44,43,45,42,41);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,20)(18,25)(19,23)(21,35)(22,33)(24,27)(26,41)(28,40)(29,30)(31,38)(32,34)(36,39)(37,47)(42,45)(43,44)(46,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,34,48,23,38,22,37)(18,26,28,42,29,43,27,36)(19,31,33,47,20,35,32,46)(24,39,25,41,40,45,30,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206485);