#115371

order	512
nilpotency class	3
isoclinism	isoclinic class of #115364
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

790 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,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=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22567);