#6506597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6413793);