#395254

order	512
nilpotency class	3
isoclinism	isoclinic class of #395254
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	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1684 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29)(16,31)(17,32)(19,34)(20,33)(22,35)(23,36)(24,38)(26,37)(28,40)(30,39)(41,44)(42,47)(43,61)(45,49)(46,55)(48,57)(50,52)(51,56)(53,59)(54,60)(58,62)(63,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,45,26,44,25,16)(5,17,28,46,30,48,29,47)(10,35,37,31,38,41,21,49)(11,32,39,42,40,57,27,55)(23,51,52,62,54,63,53,43)(36,58,59,56,60,61,50,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,37)(13,21)(14,38)(15,41)(17,43)(18,22)(19,44)(20,45)(27,50)(28,52)(29,53)(30,54)(32,56)(33,35)(34,49)(39,59)(40,60)(42,61)(46,51)(47,63)(48,62)(55,58)(57,64);(1,5,6,28,8,30,7,29)(2,11,12,39,14,40,13,27)(3,17,18,46,20,48,19,47)(4,23,24,52,26,54,25,53)(9,32,15,42,34,57,33,55)(10,36,37,59,38,60,21,50)(16,43,22,51,45,62,44,63)(31,56,41,61,49,64,35,58)

Polycyclic presentation:

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

Relationships

maximal subgroup of	#72264941, #72264954, #72265017
maximal normal subgroup of	#72264941, #72264954, #72265017
maximal quotient of	#72314091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302450);