#425013

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15327
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1047 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)(3,15,19,33)(4,10)(5,24,27,39)(6,12)(7,14)(8,13)(9,18,32,20)(11,28,40,26)(16,44,47,42)(17,48)(21,52,55,57)(22,36)(23,37)(25,43)(29,41)(30,51,34,46)(31,50)(35,54,61,56)(38,49)(45,59)(53,60)(58,63)(62,64);(1,3)(2,9)(4,16)(5,25)(6,20)(7,18)(8,19)(10,34)(11,38)(12,33)(13,32)(14,15)(17,26)(21,53)(22,51)(23,47)(24,48)(27,49)(28,50)(29,46)(30,37)(31,39)(35,60)(36,42)(40,43)(41,44)(45,54)(52,59)(55,64)(56,63)(57,58)(61,62);(1,4,6,22,8,23,7,29)(2,10,12,36,13,37,14,41)(3,16,20,51,19,47,18,46)(5,21,28,56,27,55,26,54)(9,30,33,44,32,34,15,42)(11,35,24,57,40,61,39,52)(17,45,25,53,50,63,49,64)(31,58,38,62,48,59,43,60);(1,5)(2,11)(3,17)(4,21)(6,28)(7,26)(8,27)(9,31)(10,35)(12,24)(13,40)(14,39)(15,43)(16,45)(18,49)(19,50)(20,25)(22,56)(23,55)(29,54)(30,58)(32,48)(33,38)(34,59)(36,57)(37,61)(41,52)(42,60)(44,62)(46,64)(47,63)(51,53)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332209);