#6469533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376729);