#267936

order	512
nilpotency class	3
isoclinism	isoclinic class of #267899
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

894 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:

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

Polycyclic presentation:

2^9|a^2=fh,d^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175132);