#511947

order	512
nilpotency class	4
isoclinism	isoclinic class of #511842
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#32926
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1024 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)(3,13)(4,18)(5,23)(6,11)(7,12)(8,15)(9,22)(10,25)(14,38)(16,36)(17,37)(19,48)(20,46)(21,52)(24,55)(26,34)(27,57)(28,42)(29,43)(30,60)(31,51)(32,62)(33,56)(35,41)(39,47)(40,58)(44,63)(45,49)(50,61)(53,54)(59,64);(1,3,5,15,7,17,25,43)(2,8,10,13,12,29,23,37)(4,19,21,49,20,14,39,40)(6,16,24,42,26,44,56,64)(9,30,32,61,31,27,54,35)(11,28,33,36,34,59,55,63)(18,45,47,48,46,58,52,38)(22,50,53,60,51,41,62,57);(1,4)(2,9)(3,14)(5,21)(6,22)(7,20)(8,27)(10,32)(11,18)(12,31)(13,35)(15,40)(16,41)(17,19)(23,54)(24,53)(25,39)(26,51)(28,58)(29,30)(33,47)(34,46)(36,38)(37,61)(42,57)(43,49)(44,50)(45,59)(48,63)(52,55)(56,62)(60,64);(4,20)(6,26)(9,31)(14,19)(16,44)(18,46)(21,39)(24,56)(27,30)(32,54)(35,61)(38,48)(40,49)(42,64)(45,58)(47,52)

Polycyclic presentation:

2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^c=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250025, #72250039, #72250212, …
maximal normal subgroup of	#72250025, #72250039, #72250212, …
maximal quotient of	#72250315, #72280602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419143);