#6508903

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500474
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

1774 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)(45,47)(46,48);(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)(20,35,37,46,36,30,38,47)(25,42,34,48,43,39,44,45);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20)(18,25)(19,30)(21,37)(22,38)(23,36)(24,39)(26,44)(27,34)(28,43)(29,45)(31,46)(32,47)(33,35)(40,48)(41,42);(4,12)(7,16)(9,11)(13,15)(20,36)(25,43)(30,35)(34,44)(37,38)(39,42)(45,48)(46,47);(3,10)(6,14)(9,11)(13,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,37,36,38)(24,29,41,40)(25,44,43,34)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6416099);