#273754

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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	#3518	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 180950);