#6722226

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,41)(29,45)(30,35)(31,34)(32,39)(33,42)(37,48)(38,47)(40,46)(43,49)(44,54)(50,53)(51,55)(52,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,38)(29,40)(30,39)(31,37)(33,47)(34,46)(35,36)(41,50)(42,53)(43,52)(44,51)(45,48)(49,55)(54,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,37)(28,42)(29,31)(32,46)(33,41)(34,45)(36,48)(38,52)(39,40)(47,56)(50,55)(51,53);(10,16)(14,17)(15,22)(21,23)(25,28,30,42)(26,33,35,41)(27,38,39,53)(29,43,31,44)(32,47,36,50)(34,49,45,54)(37,51,40,52)(46,55,48,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,29)(26,34)(27,39)(28,43)(30,31)(33,49)(35,45)(38,53)(41,54)(42,44)(46,48)(55,56)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=i,e^2=g,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, 6629422);