#265110

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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)(11,19,25,33)(12,26,29,36)(13,30,24,17)(14,21,15,20)(18,31,35,28)(23,37,39,32)(27,40,38,34);(1,3,6,7)(2,4,8,5)(9,11,15,13,16,25,14,24)(10,17,21,33,22,30,20,19)(12,23,31,27,29,39,28,38)(18,32,36,40,35,37,26,34);(2,3)(4,5)(7,8)(9,12)(10,18)(11,23)(13,27)(14,28)(15,31)(16,29)(17,32)(19,34)(20,26)(21,36)(22,35)(24,38)(25,39)(30,37)(33,40);(9,13,15,25,16,24,14,11)(10,19,21,17,22,33,20,30)(12,27,31,39,29,38,28,23)(18,34,36,32,35,40,26,37)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=eh,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,d^b=dh,f^b=fhi,g^b=ghi,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, 172306);