#511497

order	512
nilpotency class	4
isoclinism	isoclinic class of #511495
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32613
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

426 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:

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

Polycyclic presentation:

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

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, 418693);