#285522

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192718);