#253494

order	512
nilpotency class	3
isoclinism	isoclinic class of #253459
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	C₂³×C₄	C₂⁴
prime radical	#2404	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

861 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160690);