#253375

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	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 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)(9,10,14,18)(11,15,21,29)(12,16,22,30)(13,24,26,17)(19,32,25,27)(20,34,36,28)(23,38,37,31)(33,39,35,40);(3,8)(5,6)(11,19,21,25)(12,22)(15,27,29,32)(16,30)(20,35,36,33)(23,37)(28,40,34,39)(31,38);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,25)(14,21)(16,28)(17,32)(18,29)(19,26)(22,36)(23,35)(24,27)(30,34)(31,40)(33,37)(38,39);(9,12)(10,16)(11,20)(13,23)(14,22)(15,28)(17,31)(18,30)(19,33)(21,36)(24,38)(25,35)(26,37)(27,39)(29,34)(32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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, 160571);