#6651573

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	#30503	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(11,14)(12,15)(13,16)(17,18,23,28)(19,24,29,38)(20,26,33,42)(21,27,22,25)(30,37,43,48)(31,39,34,36)(32,40,35,41)(44,46,45,47);(1,2)(3,6)(4,5)(7,8)(11,12)(14,15)(17,19,23,29)(18,24,28,38)(20,30,33,43)(21,34,22,31)(25,39,27,36)(26,37,42,48)(32,45,35,44)(40,47,41,46);(9,11,13,12)(10,14,16,15)(18,25)(19,29)(21,22)(24,36)(26,40)(27,28)(30,43)(32,35)(37,46)(38,39)(41,42)(47,48);(2,5)(6,7)(9,12,13,11)(10,15,16,14)(17,20,23,33)(18,26,28,42)(19,30,29,43)(21,35,22,32)(24,37,38,48)(25,40,27,41)(31,45,34,44)(36,46,39,47);(1,3,4,8)(2,6,5,7)(10,16)(14,15)(17,21,23,22)(18,25,28,27)(19,31,29,34)(20,32,33,35)(24,36,38,39)(26,41,42,40)(30,44,43,45)(37,47,48,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6558769);