#300491

order	512
nilpotency class	3
isoclinism	isoclinic class of #300469
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	#62557	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=ehi,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,d^c=di,g^c=ghi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207687);