#273637

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
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	#10218	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

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, 180833);