#6755866

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

1383 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)(25,26,31,35)(27,32,38,48)(28,39,42,51)(29,34,30,44)(33,45,52,41)(36,53,40,56)(37,47,43,49)(46,54,50,55);(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)(26,32)(28,40)(29,43)(30,37)(31,38)(33,50)(34,49)(35,48)(36,42)(39,56)(41,54)(44,47)(45,55)(46,52)(51,53);(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)(29,41)(30,45)(31,42)(32,46)(34,51)(35,52)(37,54)(38,40)(39,44)(43,55)(47,53)(48,50)(49,56);(10,17)(15,19)(16,18)(23,24)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,41,42,45)(32,47,48,49)(33,51,52,39)(36,54,40,55)(46,53,50,56);(2,6)(5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,38)(28,42)(32,48)(33,52)(37,43)(39,51)(41,45)(47,49)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,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, 6663062);