#253690

order	512
nilpotency class	3
isoclinism	isoclinic class of #253651
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

704 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,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, 160886);