#405291

order	512
nilpotency class	3
isoclinism	isoclinic class of #405290
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14911
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,14)(3,15,20,32)(4,21,26,36)(5,11,29,40)(6,13,7,12)(9,19,33,18)(10,25,37,24)(16,43,46,30)(17,42,48,58)(22,34,45,41)(23,52,55,50)(27,39,28,38)(31,49,57,47)(35,53,60,54)(44,56,63,62)(51,61,64,59);(1,3)(2,9)(4,22,26,45)(5,17)(6,19)(7,18)(8,20)(10,34,37,41)(11,31)(12,32)(13,15)(14,33)(16,25,46,24)(21,30,36,43)(23,51,55,64)(27,49)(28,47)(29,48)(35,59,60,61)(38,58)(39,42)(40,57)(44,54,63,53)(50,56,52,62);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,30)(11,35)(12,36)(13,21)(14,37)(15,41)(17,44)(18,45)(19,22)(20,46)(27,54)(28,53)(29,55)(31,56)(32,34)(33,43)(38,52)(39,50)(40,60)(42,61)(47,64)(48,63)(49,51)(57,62)(58,59);(1,5)(2,11)(3,17)(4,23)(6,27)(7,28)(8,29)(9,31)(10,35)(12,38)(13,39)(14,40)(15,42)(16,44)(18,47)(19,49)(20,48)(21,50)(22,51)(24,53)(25,54)(26,55)(30,56)(32,58)(33,57)(34,59)(36,52)(37,60)(41,61)(43,62)(45,64)(46,63)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312487);