#114628

order	512
nilpotency class	3
isoclinism	isoclinic class of #114623
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

795 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:

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

Polycyclic presentation:

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

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, 21824);