#300510

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3559	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207706);