#6508911

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,42)(31,41)(33,40)(35,39)(36,44)(38,43)(45,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,41,26,40)(20,35,37,47,38,30,36,46)(25,42,34,48,44,39,43,45);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,38)(24,39)(26,43)(27,34)(28,44)(29,45)(31,47)(32,46)(33,35)(40,48)(41,42);(4,12)(7,16)(9,11)(13,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,41,40)(25,34,44,43)(30,46,35,47)(39,48,42,45);(3,10)(6,14)(9,11)(13,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,37,38,36)(24,29,41,40)(25,43,44,34)(30,47,35,46)(39,45,42,48)

Polycyclic presentation:

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

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, 6416107);