#6357920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
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	#62541	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1863 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,23,25)(19,28,31,39)(20,26,33,40)(21,27,22,36)(24,34,37,30)(29,43,46,48)(32,41,35,42)(38,44,47,45);(2,6)(7,14)(8,13)(9,15)(17,19,22,30,23,31,21,34)(18,24,27,28,25,37,36,39)(20,29,32,45,33,46,35,44)(26,38,42,43,40,47,41,48);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(18,25)(24,37)(26,40)(27,36)(28,39)(38,47)(41,42)(43,48);(3,10)(7,14)(9,15)(11,16)(17,20)(18,26)(19,29)(21,35)(22,32)(23,33)(24,38)(25,40)(27,42)(28,43)(30,45)(31,46)(34,44)(36,41)(37,47)(39,48);(17,21,23,22)(18,27,25,36)(19,30,31,34)(20,32,33,35)(24,39,37,28)(26,41,40,42)(29,44,46,45)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gh,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=di,e^d=eh

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, 6265116);