#253262

order	512
nilpotency class	3
isoclinism	isoclinic class of #253219
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	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

903 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)(4,6,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,34)(23,36)(24,30)(25,31)(29,37)(32,40)(33,39)(35,38);(2,5)(6,7)(9,11,14,21)(10,15,18,28)(12,22)(13,20,25,33)(16,26)(17,27,31,39)(19,24)(23,35)(29,38)(30,34)(32,37)(36,40);(2,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,30)(20,32)(21,22)(25,37)(27,38)(28,34)(31,36)(33,35)(39,40);(2,5)(6,7)(9,13,14,25)(10,17,18,31)(11,20,21,33)(12,23,24,37)(15,27,28,39)(16,29,30,36)(19,32,22,35)(26,38,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160458);