#6436968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,30,35)(27,31,38,46)(28,33,40,49)(29,32,42,34)(36,44,39,48)(37,43,52,45)(41,47,54,50)(51,55,53,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,30,38)(26,31,35,46)(28,39,40,36)(29,37,42,52)(32,43,34,45)(33,48,49,44)(41,53,54,51)(47,56,50,55);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(10,15)(14,20)(16,22)(21,24)(25,28,30,40)(26,33,35,49)(27,36,38,39)(29,41,42,54)(31,44,46,48)(32,47,34,50)(37,51,52,53)(43,55,45,56);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(28,40)(33,49)(36,39)(41,54)(44,48)(47,50)(51,53)(55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,g^c=gi,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, 6344164);