#268122

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
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	#61713	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 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:

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

Polycyclic presentation:

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

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, 175318);