#6571043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478239);