#6458709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,33)(21,25)(22,36)(23,27)(24,32)(26,37)(29,43)(30,41)(31,42)(34,40)(35,48)(38,45)(39,46)(44,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,32,23,31,37,46)(18,24,26,28,27,39,36,42)(20,29,34,44,21,30,35,45)(25,38,40,41,33,47,48,43);(2,5)(6,8)(11,15)(13,16)(17,20,23,21)(18,25,27,33)(19,29,31,30)(22,34,37,35)(24,38,39,47)(26,40,36,48)(28,41,42,43)(32,44,46,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,23,20)(18,25,27,33)(19,30,31,29)(22,35,37,34)(24,38,39,47)(26,40,36,48)(28,41,42,43)(32,45,46,44);(19,31)(20,21)(24,39)(25,33)(28,42)(32,46)(34,35)(40,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,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, 6365905);