#6871520

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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)(4,5)(7,8)(10,13)(11,15)(17,18,23,35,22,26,21,27)(19,24,36,46,29,39,30,38)(20,31,34,25,33,41,32,40)(28,42,45,37,43,48,44,47);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,26,39)(20,28,33,43)(21,30,23,36)(25,37,40,47)(27,38,35,46)(31,42,41,48)(32,44,34,45);(10,13)(14,16)(17,20,22,33)(18,25,26,40)(19,28,29,43)(21,34,23,32)(24,37,39,47)(27,41,35,31)(30,45,36,44)(38,48,46,42);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,26)(19,29)(25,40)(27,35)(28,43)(30,36)(31,41)(44,45);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 6778716);