#6720348

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1670 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,28)(19,24,32,39)(20,25,34,41)(21,26,22,27)(29,36,31,38)(30,37,44,47)(33,40,35,42)(43,46,45,48);(1,3)(2,5)(4,8)(6,7)(17,19,23,32)(18,24,28,39)(20,30,34,44)(21,31,22,29)(25,37,41,47)(26,38,27,36)(33,43,35,45)(40,46,42,48);(9,11,13,12)(10,14,15,16)(19,29)(21,22)(24,36)(26,27)(30,43)(31,32)(33,35)(37,46)(38,39)(40,42)(44,45)(47,48);(2,6)(5,7)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,42)(27,40)(28,41)(29,43)(31,45)(32,44)(36,46)(38,48)(39,47);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21)(18,26)(20,33)(22,23)(25,40)(27,28)(29,31)(30,44)(34,35)(36,38)(37,47)(41,42)

Polycyclic presentation:

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

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, 6627544);