#285000

order	512
nilpotency class	3
isoclinism	isoclinic class of #284987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 192196);