#113852

order	512
nilpotency class	3
isoclinism	isoclinic class of #113838
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh,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, 21048);