#6544007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1414 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451203);