#6498965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1220 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:

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

Polycyclic presentation:

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

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, 6406161);