#6557920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,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, 6465116);