#187571

order	512
nilpotency class	3
isoclinism	isoclinic class of #187442
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2152 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)(3,12,4,8)(5,15)(6,10)(7,11)(9,13,16,14)(17,18,22,28,23,26,21,27)(19,29,33,24,32,39,31,38)(20,25,34,41,35,40,36,42)(30,44,45,48,46,43,47,37);(1,3,7,4)(2,8,11,12)(5,13,6,14)(9,10,16,15)(17,19)(18,24)(20,30)(21,33)(22,31)(23,32)(25,37)(26,38)(27,39)(28,29)(34,47)(35,46)(36,45)(40,48)(41,43)(42,44);(1,4,7,3)(2,9,11,16)(5,13,6,14)(8,10,12,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,40)(27,42)(28,41)(29,43)(31,47)(32,46)(33,45)(38,48)(39,44);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,16,12,9)(18,26)(24,38)(25,40)(27,28)(29,39)(37,48)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,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, 94767);