#6825380

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,29,37)(20,30,32,25)(21,26,35,36)(27,41,38,31)(28,34,40,47)(33,43,45,39)(42,48,46,44);(2,5)(3,7)(17,19,22,29)(18,24,23,37)(20,31,32,41)(21,34,35,47)(25,38,30,27)(26,40,36,28)(33,46,45,42)(39,48,43,44);(10,13)(14,16)(17,20,18,25,22,32,23,30)(19,27,37,41,29,38,24,31)(21,33,26,39,35,45,36,43)(28,42,47,48,40,46,34,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(20,32)(25,30)(27,38)(31,41)(33,45)(39,43)(42,46)(44,48);(1,3,4,7)(2,6,5,8)(17,21,18,26,22,35,23,36)(19,28,37,47,29,40,24,34)(20,33,25,39,32,45,30,43)(27,42,41,48,38,46,31,44)

Polycyclic presentation:

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

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, 6732576);