#6458521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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)(19,28)(20,25)(21,36)(22,27)(23,31)(24,34)(26,29)(30,43)(32,44)(33,42)(35,40)(37,48)(38,47)(39,45)(41,46);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,29,45,22,32,21,31)(18,23,36,44,27,39,26,28)(20,33,46,47,34,30,35,48)(24,38,41,42,25,37,40,43);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,24)(19,30)(21,35)(22,34)(23,37)(25,27)(26,40)(28,42)(29,46)(31,47)(32,33)(36,41)(38,39)(43,44)(45,48);(11,15)(13,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,38)(24,27)(26,41)(28,43)(29,46)(31,47)(32,33)(36,40)(37,39)(42,44)(45,48);(20,34)(24,25)(30,33)(35,46)(37,38)(40,41)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=gh,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^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, 6365717);