#253382

order	512
nilpotency class	3
isoclinism	isoclinic class of #253363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18152	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,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, 160578);