#6357649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357620
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3508	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1790 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)(3,5)(4,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,18,23,27)(19,28,29,42)(20,25,34,41)(21,26,22,36)(24,31,37,32)(30,43,44,48)(33,39,35,40)(38,46,47,45);(1,3,6,4,8,14,7,13)(2,9,11,5,12,15,10,16)(17,19,22,32,23,29,21,31)(18,24,26,28,27,37,36,42)(20,30,33,45,34,44,35,46)(25,38,40,43,41,47,39,48);(1,4,6,14,8,13,7,3)(2,5,11,15,12,16,10,9)(19,29)(24,37)(28,42)(30,44)(31,32)(38,47)(43,48)(45,46);(1,5)(2,3)(4,11)(6,15)(7,9)(8,16)(10,13)(12,14)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,38)(26,40)(27,41)(28,43)(29,44)(31,46)(32,45)(36,39)(37,47)(42,48);(1,6,8,7)(2,10,12,11)(3,4,14,13)(5,9,16,15)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,33,34,35)(24,28,37,42)(25,39,41,40)(30,45,44,46)(38,48,47,43)

Polycyclic presentation:

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

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, 6264845);