#6489207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	#25733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1858 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,23,27,24,28,22,37)(19,29,32,41,33,40,38,25)(20,34,35,43,21,26,36,44)(30,46,48,39,31,45,47,42);(11,12)(14,15)(17,19,24,33)(18,25,28,41)(20,31,21,30)(22,32,23,38)(26,42,34,39)(27,40,37,29)(35,48,36,47)(43,46,44,45);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,24,21)(18,26,28,34)(19,30,33,31)(22,36,23,35)(25,39,41,42)(27,44,37,43)(29,45,40,46)(32,48,38,47);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,26,28,34)(19,31,33,30)(22,35,23,36)(25,39,41,42)(27,44,37,43)(29,45,40,46)(32,47,38,48);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,22,24,23)(18,27,28,37)(19,32,33,38)(20,35,21,36)(25,40,41,29)(26,43,34,44)(30,47,31,48)(39,45,42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396403);