#401793

order	512
nilpotency class	3
isoclinism	isoclinic class of #401792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14823
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 308989);