#273722

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

766 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)(8,11,16,15)(17,18,23,27,24,21,22,39)(19,28,31,25,33,42,32,30)(20,34,37,36,38,44,40,26)(29,46,47,43,48,41,35,45);(1,2,6,10)(3,8,13,16)(4,9,7,5)(11,12,15,14)(17,19)(18,25)(20,35)(21,30)(22,32)(23,31)(24,33)(26,43)(27,42)(28,39)(29,37)(34,41)(36,45)(38,47)(40,48)(44,46);(1,3,6,13)(2,8,10,16)(4,11,7,15)(5,14,9,12)(17,20)(18,26)(19,29)(21,36)(22,37)(23,40)(24,38)(25,41)(27,44)(28,45)(30,46)(31,35)(32,47)(33,48)(34,39)(42,43);(1,4)(2,9)(3,11)(5,10)(6,7)(8,12)(13,15)(14,16)(17,21,24,18)(19,30,33,25)(20,36,38,26)(22,27,23,39)(28,32,42,31)(29,46,48,41)(34,37,44,40)(35,45,47,43)

Polycyclic presentation:

2^9|a^2=f,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, 180918);