#253499

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=i,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, 160695);