#6458711

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

1572 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)(19,29)(20,34)(21,26)(22,27)(23,38)(24,28)(25,32)(30,45)(31,44)(33,40)(35,43)(36,42)(37,41)(39,46)(47,48);(9,11)(10,13)(12,15)(14,16)(17,19,23,32,24,33,22,37)(18,25,27,29,28,41,38,40)(20,30,35,47,21,31,36,46)(26,39,43,44,34,48,42,45);(2,5)(6,8)(11,15)(13,16)(17,20,24,21)(18,26,28,34)(19,30,33,31)(22,36,23,35)(25,39,41,48)(27,43,38,42)(29,44,40,45)(32,47,37,46);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,24,20)(18,26,28,34)(19,31,33,30)(22,35,23,36)(25,39,41,48)(27,43,38,42)(29,44,40,45)(32,46,37,47);(17,22,24,23)(18,27,28,38)(19,32,33,37)(20,35,21,36)(25,40,41,29)(26,42,34,43)(30,46,31,47)(39,44,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^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,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, 6365907);