#6351566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351559
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3545	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,31)(25,35)(30,38)(32,41)(34,42)(36,39)(37,40)(43,46)(44,48)(45,47);(4,13)(7,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,39)(27,41)(28,29)(33,47)(34,44)(35,46)(37,45)(40,48)(42,43);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,20)(18,25)(19,30)(21,35)(22,37)(23,34)(24,38)(26,40)(27,33)(28,42)(29,43)(31,45)(32,44)(36,46)(39,48)(41,47);(20,34)(25,40)(30,44)(33,42)(35,37)(38,48)(43,47)(45,46);(18,26)(24,39)(25,40)(27,28)(29,41)(33,42)(38,48)(43,47)

Polycyclic presentation:

2^9|a^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279356, #72279869, #72280141, …
maximal normal subgroup of	#72279356, #72279869, #72280141, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258762);