#115127

order	512
nilpotency class	3
isoclinism	isoclinic class of #115124
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

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

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=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ehi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22323);