#300505

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	#62436	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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)(3,10)(4,14)(5,8)(6,9)(7,13)(11,12)(15,16)(17,18)(19,27)(20,31)(21,36)(22,26)(23,29)(24,34)(25,33)(28,42)(30,41)(32,45)(35,46)(37,44)(38,39)(40,43)(47,48);(1,2)(3,11)(4,8)(5,14)(6,9)(7,13)(10,12)(15,16)(17,19,21,29,22,30,33,39)(18,23,25,27,26,38,36,41)(20,28,34,43,35,44,45,47)(24,37,31,40,32,42,46,48);(1,3,4,13,6,12,5,15)(2,7,8,10,9,16,14,11)(17,18)(19,23,30,38)(20,32)(21,25)(22,26)(24,35)(27,39,41,29)(28,42,44,37)(31,45)(33,36)(34,46)(40,43,48,47);(3,12)(7,16)(10,11)(13,15)(17,20,33,45,22,35,21,34)(18,24,36,46,26,32,25,31)(19,28,39,47,30,44,29,43)(23,37,41,48,38,42,27,40)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=eh,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, 207701);