#6355994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355925
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	C₄²	C₂⁵
prime radical	#61970	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1503 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263190);