#199036

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1047 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)(3,5)(4,12,14,8)(6,11)(7,9)(10,13,16,15)(17,18,22,28,23,26,21,27)(19,29,33,24,31,39,32,40)(20,34,36,25,37,42,35,41)(30,38,46,43,44,48,45,47);(1,3,5,2,7,11,6,9)(4,10,13,8,14,16,15,12)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,40)(27,39)(28,29)(34,43)(35,46)(36,45)(37,44)(41,48)(42,47);(1,4)(2,8)(3,10)(5,13)(6,15)(7,14)(9,12)(11,16)(17,20,23,37)(18,25,26,41)(19,30,31,44)(21,36,22,35)(24,38,40,48)(27,42,28,34)(29,43,39,47)(32,46,33,45);(1,5,7,6)(2,3,9,11)(4,13,14,15)(8,10,12,16)(18,26)(19,31)(25,41)(27,28)(30,44)(32,33)(34,42)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

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, 106232);