#273737

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	#62239	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

785 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:

(2,6,8,11)(3,9)(7,14,16,15)(10,12)(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)(3,7)(4,11)(5,8)(6,13)(9,16)(10,14)(12,15)(17,19,23,34)(18,24,28,43)(20,31,37,47)(21,33,22,32)(25,44,26,41)(27,42,38,29)(30,39,48,36)(35,40,45,46);(1,3,5,9)(2,7,8,16)(4,12,13,10)(6,14,11,15)(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);(1,2)(3,7)(4,11)(5,8)(6,13)(9,16)(10,14)(12,15)(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=h,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, 180933);