#267888

order	512
nilpotency class	3
isoclinism	isoclinic class of #267851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60565	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175084);