#253500

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,13)(3,9,17,34)(4,19,22,50)(5,11,26,42)(6,27,29,41)(8,14,25,12)(10,35,37,23)(15,44,20,52)(16,31,49,61)(18,33,48,43)(21,39,56,38)(24,40,53,51)(28,32,58,62)(30,54,60,45)(36,57,63,47)(46,59,55,64);(1,3,8,18)(2,9,14,33)(4,20,53,46)(5,16,6,28)(7,17,25,48)(10,36,38,60)(11,31,27,32)(12,43,13,34)(15,24,55,22)(19,44,51,64)(21,54,35,47)(23,57,56,45)(26,49,29,58)(30,37,63,39)(40,59,50,52)(41,62,42,61);(1,4,8,24)(2,10,14,39)(3,15,18,46)(5,23,6,21)(7,22,25,53)(9,30,33,36)(11,40,27,19)(12,38,13,37)(16,47,28,45)(17,20,48,55)(26,35,29,56)(31,44,32,59)(34,60,43,63)(41,50,42,51)(49,57,58,54)(52,62,64,61);(1,5,25,29)(2,11,12,41)(3,16,48,58)(4,21,53,35)(6,7,26,8)(9,31,43,62)(10,19,38,51)(13,42,14,27)(15,45,55,57)(17,49,18,28)(20,54,46,47)(22,56,24,23)(30,59,63,52)(32,34,61,33)(36,44,60,64)(37,50,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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, 160696);