#274073

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181269);