#395250

order	512
nilpotency class	3
isoclinism	isoclinic class of #395250
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14722
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1690 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)(3,15)(4,10)(5,26)(6,13)(7,12)(8,14)(9,18)(11,29)(16,41)(17,31)(19,33)(20,32)(21,30)(22,50)(23,40)(24,36)(25,37)(27,39)(28,38)(34,44)(35,51)(42,47)(43,62)(45,54)(46,58)(48,57)(49,61)(52,55)(53,60)(56,63)(59,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,32)(4,21,23,44,25,48,24,16)(5,17,27,45,28,46,29,47)(10,34,36,30,37,41,40,57)(11,31,38,42,39,58,26,54)(22,49,51,63,52,64,53,43)(35,59,50,56,60,61,55,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30)(11,35)(12,36)(13,40)(14,37)(15,41)(17,43)(18,21)(19,48)(20,44)(26,55)(27,51)(28,52)(29,53)(31,56)(32,34)(33,57)(38,50)(39,60)(42,61)(45,49)(46,63)(47,64)(54,59)(58,62);(1,5,6,27,8,28,7,29)(2,11,12,38,14,39,13,26)(3,17,18,45,20,46,19,47)(4,22,23,51,25,52,24,53)(9,31,15,42,33,58,32,54)(10,35,36,50,37,60,40,55)(16,43,21,49,44,63,48,64)(30,56,41,61,57,62,34,59)

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^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,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264906, #72264954, #72264996
maximal normal subgroup of	#72264906, #72264954, #72264996
maximal quotient of	#72265182

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302446);