#273514

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
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	#61081	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1029 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180710);