#6710073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1236 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)(28,39)(29,33)(30,34)(31,38)(32,41)(36,46)(37,47)(40,49)(42,52)(43,51)(44,48)(45,50)(53,54)(55,56);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,40)(29,38)(30,44)(32,49)(33,35)(34,47)(36,42)(37,43)(39,56)(41,55)(45,54)(46,50)(48,51)(52,53);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,36)(29,41)(30,45)(31,46)(33,39)(34,52)(35,53)(37,55)(38,54)(40,44)(42,43)(47,49)(48,56)(50,51);(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)(28,41)(31,35)(34,51)(36,54)(40,55)(42,45)(47,48);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(28,41)(30,43)(31,47)(35,48)(36,40)(38,44)(46,56)(49,53)(50,52)(54,55)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,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, 6617269);