#6350778

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60930	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257974);