#115057

order	512
nilpotency class	3
isoclinism	isoclinic class of #115056
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60799	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dhi,e^c=ehi,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, 22253);