#115595

order	512
nilpotency class	3
isoclinism	isoclinic class of #10192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

777 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22791);