#232914

order	512
nilpotency class	3
isoclinism	isoclinic class of #232905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi

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, 140110);