#6860509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6767705);