#6496140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
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	C₂²×C₄	C₂⁵
prime radical	#62861	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403336);