#499025

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2033
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1039 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)(3,14)(4,20)(5,10)(6,28)(7,13)(8,17)(9,23)(11,27)(12,26)(15,44)(16,34)(18,32)(19,33)(21,55)(22,53)(24,54)(25,40)(29,36)(30,39)(31,61)(35,50)(37,58)(38,57)(41,59)(42,47)(43,48)(45,52)(46,62)(49,60)(51,63)(56,64);(1,3)(2,8)(4,21)(5,16)(6,18)(7,19)(9,35)(10,14)(11,33)(12,32)(13,34)(15,24)(17,27)(20,52)(22,46)(23,56)(25,45)(26,48)(28,60)(29,61)(30,51)(31,38)(36,64)(37,42)(39,43)(40,44)(41,63)(47,57)(49,59)(50,58)(53,55)(54,62);(1,4,5,22,7,24,27,25)(2,9,11,37,13,38,10,36)(3,15,17,46,19,21,16,45)(6,29,26,57,30,58,59,23)(8,31,14,42,34,35,33,64)(12,40,28,54,41,53,39,20)(18,50,49,47,51,61,48,56)(32,55,43,62,63,44,60,52);(1,5,7,27)(2,10,13,11)(3,16,19,17)(4,22,24,25)(6,26,30,59)(8,14,34,33)(9,36,38,37)(12,39,41,28)(15,45,21,46)(18,48,51,49)(20,53,54,40)(23,29,57,58)(31,42,35,64)(32,43,63,60)(44,52,55,62)(47,50,56,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320602, #72320647, #72320745
maximal normal subgroup of	#72320602, #72320647, #72320745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406221);