#6825428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(17,18,23,22)(19,24,29,37)(20,25,33,32)(21,26,35,36)(27,30,38,43)(28,34,40,47)(31,39,45,46)(41,42,44,48);(2,5)(3,7)(17,19,23,29)(18,24,22,37)(20,30,33,43)(21,34,35,47)(25,38,32,27)(26,40,36,28)(31,44,45,41)(39,48,46,42);(10,13)(14,16)(17,20,18,25,23,33,22,32)(19,27,37,43,29,38,24,30)(21,31,26,39,35,45,36,46)(28,41,47,48,40,44,34,42);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(21,35)(26,36)(28,40)(31,45)(34,47)(39,46)(41,44)(42,48);(1,3,4,7)(2,6,5,8)(17,21,18,26,23,35,22,36)(19,28,37,47,29,40,24,34)(20,31,25,39,33,45,32,46)(27,41,43,48,38,44,30,42)

Polycyclic presentation:

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

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, 6732624);