#6519473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37544	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 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)(3,6,7,8)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,42)(37,41)(43,45)(44,48)(46,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,33,22,32)(18,24,27,39,28,40,26,29)(20,30,36,45,35,44,37,46)(25,38,34,47,41,48,42,43);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,37,23,35,21,36)(18,25,26,42,28,41,27,34)(19,30,32,46,33,44,31,45)(24,38,29,43,40,48,39,47);(9,11,12,15)(10,14,13,16)(20,35)(25,41)(30,44)(34,42)(36,37)(38,48)(43,47)(45,46);(10,13)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,35,37)(24,29,40,39)(25,42,41,34)(30,45,44,46)(38,43,48,47)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6426669);