#6479213

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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

1354 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)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,32)(28,40)(29,34)(30,43)(31,35)(33,41)(36,50)(37,46)(38,47)(39,53)(42,48)(44,49)(45,51)(52,55)(54,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,38)(26,32,35,47)(28,36,42,52)(29,37,30,39)(33,45,49,56)(34,46,43,53)(40,50,48,55)(41,51,44,54);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,36)(29,41)(30,44)(31,42)(32,45)(34,48)(35,49)(37,51)(38,52)(39,54)(40,43)(46,55)(47,56)(50,53);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,43)(27,37,38,39)(28,41,42,44)(32,46,47,53)(33,48,49,40)(36,51,52,54)(45,55,56,50);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(27,38)(28,42)(32,47)(33,49)(37,39)(40,48)(41,44)(46,53)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6386409);