#499047

order	512
nilpotency class	4
isoclinism	isoclinic class of #498800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2034
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1061 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)(4,21)(5,13)(6,28)(7,11)(8,14)(9,18)(10,24)(12,30)(16,44)(17,35)(19,34)(20,33)(22,54)(23,40)(25,52)(26,53)(27,41)(29,37)(31,39)(32,59)(36,47)(38,56)(42,55)(43,51)(45,63)(46,61)(48,62)(49,57)(50,60)(58,64);(1,3,7,20,8,17,5,18)(2,9,13,35,14,33,11,15)(4,22,23,16,26,46,25,45)(6,19,30,51,31,48,27,49)(10,36,37,60,38,64,42,32)(12,34,28,57,41,62,39,43)(21,44,52,54,53,63,40,61)(24,50,55,47,56,59,29,58);(1,4,5,23,8,26,7,25)(2,10,11,37,14,38,13,42)(3,16,18,46,17,45,20,22)(6,29,27,24,31,55,30,56)(9,32,15,36,33,60,35,64)(12,40,39,21,41,52,28,53)(19,50,49,47,48,59,51,58)(34,63,43,61,62,44,57,54);(3,17)(9,33)(15,35)(16,45)(18,20)(19,48)(22,46)(32,60)(34,62)(36,64)(43,57)(44,63)(47,58)(49,51)(50,59)(54,61)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,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	#72249825, #72249831, #72249832, …
maximal normal subgroup of	#72249825, #72249831, #72249832, …
maximal quotient of	#72257093, #72257129, #72322890, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406243);