#268306

order	512
nilpotency class	3
isoclinism	isoclinic class of #268283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61944	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175502);