#6649065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6556261);