#115036

order	512
nilpotency class	3
isoclinism	isoclinic class of #115035
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18)(19,27,30,23)(20,31,32,41)(21,26)(22,25)(24,33,39,34)(28,43)(29,42,47,37)(35,40)(36,45)(38,46)(44,48);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,22,30)(18,23,25,27)(20,28,32,44)(21,29,35,47)(24,38,39,36)(26,37,40,42)(31,43,41,48)(33,45,34,46);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,20)(18,24)(19,28)(21,34)(22,32)(23,36)(25,39)(26,41)(27,38)(29,46)(30,44)(31,40)(33,35)(37,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,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, 22232);