#253373

order	512
nilpotency class	3
isoclinism	isoclinic class of #253363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18334	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eh,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fi,g^c=gh,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, 160569);