#6805888

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18,22,26)(19,23,28,36)(20,30,31,45)(21,33,34,25)(24,38,39,32)(27,40,42,48)(29,44,43,37)(35,46,47,41);(1,3,6,2)(4,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,26,36)(20,27,31,42)(21,29,34,43)(24,35,39,47)(25,37,33,44)(30,40,45,48)(32,41,38,46);(10,13)(14,16)(17,20,21,32,22,31,34,38)(18,24,25,30,26,39,33,45)(19,27,29,41,28,42,43,46)(23,35,37,40,36,47,44,48);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(19,28)(20,31)(23,36)(24,39)(29,43)(30,45)(32,38)(37,44);(1,5,6,4)(2,7,3,8)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=g,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6713084);