#6529633

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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	#60919	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2670 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=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, 6436829);