#6532058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(3,5)(4,6)(7,8)(25,26,30,34)(27,35,36,48)(28,32,39,46)(29,41,43,33)(31,42,44,38)(37,49,50,56)(40,53,52,47)(45,54,55,51);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,37)(29,42)(30,36)(32,45)(33,48)(34,44)(35,41)(38,43)(39,50)(40,54)(46,55)(47,56)(49,53)(51,52);(2,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,36)(31,44)(35,48)(37,50)(38,42)(45,55)(49,56)(51,54);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,32)(27,37)(29,40)(30,39)(31,45)(33,47)(34,46)(35,49)(36,50)(38,51)(41,53)(42,54)(43,52)(44,55)(48,56);(10,16)(14,17)(15,24)(22,23)(27,36)(28,39)(31,44)(32,46)(35,48)(38,42)(40,52)(47,53)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439254);