#6462325

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(17,18,21,26,22,27,36,44)(19,28,30,39,31,43,35,23)(20,25,34,42,33,24,40,41)(29,46,47,37,32,45,48,38);(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,23,27,39)(20,32,33,29)(21,35,36,30)(24,38,25,37)(26,43,44,28)(34,47,40,48)(41,45,42,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,22,33)(18,24,27,25)(19,29,31,32)(21,34,36,40)(23,37,39,38)(26,41,44,42)(28,45,43,46)(30,47,35,48);(10,14)(13,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,38)(24,27)(26,42)(28,46)(30,47)(31,32)(35,48)(36,40)(37,39)(41,44)(43,45);(1,3,4,7)(2,6,5,8)(20,33)(24,25)(29,32)(34,40)(37,38)(41,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6369521);