#6356502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356405
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	#60988	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1504 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)(3,8)(4,6)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,30)(29,42)(31,38)(32,40)(33,41)(34,46)(35,39)(37,44)(43,48)(45,47);(2,6)(7,9)(8,13)(14,15)(17,19,22,30,23,31,21,35)(18,24,26,28,27,39,36,38)(20,29,34,44,32,43,33,45)(25,37,41,42,40,47,46,48);(2,7)(5,12)(6,9)(8,14)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,37)(26,41)(27,40)(28,42)(30,44)(31,43)(35,45)(36,46)(38,48)(39,47);(1,3,4,10)(2,8,6,13)(5,11,12,16)(7,14,9,15)(20,32)(25,40)(29,43)(33,34)(37,47)(41,46)(42,48)(44,45);(3,10)(8,13)(11,16)(14,15)(17,21,23,22)(18,26,27,36)(19,30,31,35)(20,33,32,34)(24,38,39,28)(25,41,40,46)(29,44,43,45)(37,48,47,42)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263698);