#6657496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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)(3,7,8,6)(11,15)(14,16)(17,18)(19,23)(20,30)(21,34)(22,25)(24,33)(26,35)(27,41)(28,38)(29,44)(31,45)(32,46)(36,42)(37,43)(39,47)(40,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,25,38)(20,31,32,27)(21,29,35,43)(24,39,40,36)(26,37,34,44)(30,45,46,41)(33,47,48,42);(2,5)(6,7)(10,13)(14,16)(17,20,21,33,22,32,35,48)(18,24,26,30,25,40,34,46)(19,27,29,42,28,31,43,47)(23,36,37,41,38,39,44,45);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,32)(33,42)(34,44)(35,43)(39,40)(45,46)(47,48);(9,11,12,15)(10,14,13,16)(18,25)(19,28)(20,32)(26,34)(29,43)(33,48)(36,39)(41,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei,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, 6564692);