#285412

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192608);