#253536

order	512
nilpotency class	3
isoclinism	isoclinic class of #253507
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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 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)(3,6)(4,12)(5,11)(7,9)(8,16)(10,14)(13,15)(17,18,22,26)(19,23,29,37)(20,24,30,38)(21,32,34,25)(27,40,33,35)(28,42,43,36)(31,45,44,39)(41,47,46,48);(1,3,6,2)(4,13,15,12)(5,11,7,9)(8,14,10,16)(19,27,29,33)(23,35,37,40)(28,41,43,46)(36,47,42,48);(1,4)(2,8)(3,10)(5,14)(6,15)(7,16)(9,13)(11,12)(17,19)(18,23)(20,28)(21,33)(22,29)(24,36)(25,40)(26,37)(27,34)(30,43)(31,46)(32,35)(38,42)(39,48)(41,44)(45,47);(1,5,6,7)(2,9,3,11)(4,14,15,16)(8,13,10,12)(17,20,22,30)(18,24,26,38)(19,28,29,43)(21,31,34,44)(23,36,37,42)(25,39,32,45)(27,41,33,46)(35,47,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160732);