#6755854

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26)(27,32)(28,39)(29,34)(30,43)(31,35)(33,40)(36,51)(37,46)(38,48)(41,49)(42,47)(44,50)(45,54)(52,55)(53,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,31,38)(26,32,35,48)(28,36,41,53)(29,42,30,37)(33,45,50,55)(34,47,43,46)(39,51,49,56)(40,54,44,52);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,36)(29,40)(30,44)(31,41)(32,45)(34,49)(35,50)(37,52)(38,53)(39,43)(42,54)(46,51)(47,56)(48,55);(10,17)(15,19)(16,18)(23,24)(25,29,31,30)(26,34,35,43)(27,37,38,42)(28,40,41,44)(32,46,48,47)(33,49,50,39)(36,52,53,54)(45,51,55,56);(2,6)(5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(28,41)(33,50)(36,53)(39,49)(40,44)(45,55)(51,56)(52,54)

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,d^2=i,e^2=f,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, 6663050);