#6458519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1935 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365715);