#273753

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	#62238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

786 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)(17,18,21,38,23,28,22,27)(19,29,33,43,34,42,32,24)(20,35,39,25,37,45,36,26)(30,46,31,41,48,40,47,44);(1,2,6,10)(3,8,11,16)(4,9,7,5)(12,13,15,14)(17,19)(18,24)(20,31)(21,33)(22,32)(23,34)(25,44)(26,41)(27,42)(28,43)(29,38)(30,36)(35,40)(37,47)(39,48)(45,46);(1,3,6,11)(2,8,10,16)(4,12,7,15)(5,14,9,13)(17,20)(18,25)(19,30)(21,36)(22,39)(23,37)(24,40)(26,28)(27,45)(29,41)(31,32)(33,47)(34,48)(35,38)(42,44)(43,46);(1,4)(2,9)(3,12)(5,10)(6,7)(8,13)(11,15)(14,16)(17,20)(18,26)(19,31)(21,39)(22,36)(23,37)(24,41)(25,28)(27,45)(29,40)(30,32)(33,48)(34,47)(35,38)(42,46)(43,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,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, 180949);