#6357545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1885 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=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, 6264741);