#273197

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
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	#18428	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180393);