#274012

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#61095	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=h,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,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, 181208);