#6709887

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,53)(37,51)(39,56)(40,48)(41,49)(42,50)(43,46)(44,52)(45,54)(47,55);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,29,38)(26,31,33,35)(28,39,40,55)(30,43,42,37)(32,47,48,56)(34,51,50,46)(36,44,54,41)(45,52,53,49);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,40)(26,32,33,48)(27,36,38,54)(30,44,42,41)(31,45,35,53)(34,52,50,49)(37,55,43,39)(46,56,51,47);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(26,33)(28,40)(31,35)(34,50)(36,54)(39,55)(41,44)(46,51);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,37)(30,42)(31,46)(34,50)(35,51)(36,55)(38,43)(39,54)(41,44)(45,56)(47,53)(49,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6617083);