#147946

order	512
nilpotency class	4
isoclinism	isoclinic class of #140776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11022
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

955 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307185, #72307235, #72307268, …
maximal normal subgroup of	#72307185, #72307235, #72307268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55142);