#6651426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6558622);