#6354072

order	512
nilpotency class	3
isoclinism	isoclinic class of #19368
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#60949	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1625 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,41)(30,42)(31,38)(33,39)(34,40)(35,46)(37,44)(43,47)(45,48);(2,7)(8,15)(9,10)(11,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,38,36,42)(20,29,35,44,33,45,34,43)(25,37,40,41,39,47,46,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,45)(31,43)(32,44)(36,46)(38,47)(42,48);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(19,30)(20,33)(24,38)(25,39)(28,42)(31,32)(34,35)(40,46);(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,43,45,44)(37,41,47,48)

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,d^a=di,f^a=fh,g^a=gi,d^b=dh,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, 6261268);