#468072

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16600
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1073 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)(3,13)(4,18)(5,10)(6,25)(7,12)(8,16)(9,21)(11,28)(14,43)(15,47)(17,42)(19,54)(20,34)(22,35)(23,39)(24,37)(26,38)(27,36)(29,45)(30,49)(31,53)(32,51)(33,55)(40,46)(41,50)(44,58)(48,57)(52,59)(56,61)(60,64)(62,63);(1,3)(2,8,12,31)(4,14)(5,15)(6,16)(7,17)(9,32,18,29)(10,30,37,59)(11,13,25,42)(19,44)(20,45)(21,46)(22,48)(23,49)(24,50)(26,51)(27,52)(28,53)(33,60,54,57)(34,40,38,43)(35,62,61,58)(36,47,39,41)(55,63)(56,64);(1,4)(2,9,12,18)(3,14)(5,22)(6,26)(7,21)(8,29,31,32)(10,35,37,61)(11,38,25,34)(13,40,42,43)(15,48)(16,51)(17,46)(19,23)(20,28)(24,56)(27,55)(30,58,59,62)(33,39,54,36)(41,60,47,57)(44,49)(45,53)(50,64)(52,63);(1,5)(2,10)(3,15)(4,19)(6,27)(7,24)(8,30)(9,33)(11,39)(12,37)(13,41)(14,44)(16,52)(17,50)(18,54)(20,56)(21,55)(22,26)(23,28)(25,36)(29,57)(31,59)(32,60)(34,61)(35,38)(40,58)(42,47)(43,62)(45,64)(46,63)(48,51)(49,53)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375268);