#265080

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18756	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

908 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)(3,4)(5,7)(6,8)(9,10,16,22)(11,23,26,19)(12,27,30,38)(13,17,25,34)(14,21,15,20)(18,32,36,29)(24,39,40,33)(28,37,31,35);(1,3,6,7)(2,4,8,5)(9,11,15,25,16,26,14,13)(10,17,21,19,22,34,20,23)(12,24,32,28,30,40,29,31)(18,33,38,37,36,39,27,35);(2,3)(4,5)(7,8)(9,12)(10,18)(11,24)(13,31)(14,29)(15,32)(16,30)(17,33)(19,37)(20,27)(21,38)(22,36)(23,35)(25,28)(26,40)(34,39);(1,4,6,5)(2,3,8,7)(9,13,14,26,16,25,15,11)(10,19,20,17,22,23,21,34)(12,28,29,24,30,31,32,40)(18,35,27,39,36,37,38,33)

Polycyclic presentation:

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

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, 172276);