#115071

order	512
nilpotency class	3
isoclinism	isoclinic class of #115065
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

769 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=ehi,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, 22267);