#6718029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#30251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(9,10,13,15)(11,14,12,16)(19,28)(23,37)(29,41)(30,42)(31,36)(38,46)(43,44)(45,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,26)(19,29,28,41)(20,32,33,24)(22,25,35,27)(23,36,37,31)(30,44,42,43)(34,39,47,40)(38,45,46,48);(10,15)(14,16)(17,19,21,28)(18,23,26,37)(20,30,33,42)(22,36,35,31)(24,38,32,46)(25,29,27,41)(34,48,47,45)(39,44,40,43);(2,6)(5,7)(17,20)(18,24)(19,30)(21,33)(22,34)(23,38)(25,40)(26,32)(27,39)(28,42)(29,43)(31,45)(35,47)(36,48)(37,46)(41,44);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(18,25)(20,33)(22,35)(23,29)(24,39)(26,27)(30,42)(31,36)(32,40)(37,41)(38,44)(43,46)

Polycyclic presentation:

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

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, 6625225);