#427272

order	512
nilpotency class	3
isoclinism	isoclinic class of #427271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15368
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 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)(3,15,19,34)(4,10,25,39)(5,26,29,42)(6,13,7,12)(9,20,33,18)(11,30,41,28)(16,35,46,43)(17,47,48,32)(21,31,45,57)(22,51,54,60)(23,38,24,37)(27,40,49,58)(36,53,61,52)(44,56,63,64)(50,62,55,59);(1,3)(2,9)(4,21)(5,27)(6,20)(7,18)(8,19)(10,35)(11,40)(12,34)(13,15)(14,33)(16,24)(17,28)(22,44)(23,46)(25,45)(26,32)(29,49)(30,48)(31,38)(36,56)(37,57)(39,43)(41,58)(42,47)(50,52)(51,59)(53,55)(54,63)(60,62)(61,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,38)(14,39)(15,43)(17,44)(18,45)(19,46)(20,21)(26,51)(27,55)(28,52)(29,54)(30,53)(32,56)(33,57)(34,35)(40,62)(41,61)(42,60)(47,64)(48,63)(49,50)(58,59);(1,5)(2,11)(3,17,19,48)(4,22)(6,28)(7,30)(8,29)(9,32,33,47)(10,36)(12,26)(13,42)(14,41)(15,40,34,58)(16,44,46,63)(18,27,20,49)(21,50,45,55)(23,52)(24,53)(25,54)(31,56,57,64)(35,59,43,62)(37,51)(38,60)(39,61)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264934, #72264996
maximal normal subgroup of	#72264914, #72264934, #72264996
maximal quotient of	#72265526

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334468);