#6458509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365705);