#115077

order	512
nilpotency class	3
isoclinism	isoclinic class of #115074
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

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22273);