#508111

order	512
nilpotency class	4
isoclinism	isoclinic class of #508110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29636
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3027	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

792 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)(3,14,19,41)(4,20,24,39)(5,12,26,10)(6,27,30,11)(8,28,33,17)(9,34,38,23)(15,47,51,64)(16,52,18,46)(21,37,58,42)(22,59,29,36)(25,40,60,43)(31,48,35,63)(32,54,44,56)(45,53,61,50)(49,57,55,62);(1,3)(2,8)(4,15)(5,16)(6,28)(7,19)(9,35)(10,32)(11,41)(12,44)(13,33)(14,27)(17,30)(18,26)(20,57)(21,48)(22,55)(23,50)(24,51)(25,56)(29,49)(31,38)(34,53)(36,64)(37,61)(39,62)(40,52)(42,45)(43,46)(47,59)(54,60)(58,63);(1,4)(2,9)(3,15)(5,23)(6,29)(7,24)(8,31)(10,39)(11,42)(12,20)(13,38)(14,45)(16,53)(17,55)(18,50)(19,51)(21,25)(22,30)(26,34)(27,37)(28,49)(32,62)(33,35)(36,40)(41,61)(43,59)(44,57)(46,47)(48,54)(52,64)(56,63)(58,60);(1,5,7,26)(2,10,13,12)(3,16,19,18)(4,21)(6,25,30,60)(8,32,33,44)(9,36)(11,40,27,43)(14,46,41,52)(15,48)(17,54,28,56)(20,42)(22,23)(24,58)(29,34)(31,47)(35,64)(37,39)(38,59)(45,57)(49,53)(50,55)(51,63)(61,62)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415307);