#427308

order	512
nilpotency class	3
isoclinism	isoclinic class of #427307
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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)(3,15,19,33)(4,10,23,36)(5,24,27,40)(6,13,7,12)(9,20,32,18)(11,29,38,26)(16,42,45,41)(17,46,47,31)(21,50,53,60)(22,39,28,35)(25,37,49,58)(30,48,57,44)(34,52,61,54)(43,56,63,64)(51,62,55,59);(1,3)(2,9)(4,16)(5,25)(6,20)(7,18)(8,19)(10,30)(11,37)(12,33)(13,15)(14,32)(17,26)(21,51)(22,48)(23,45)(24,31)(27,49)(28,44)(29,47)(34,59)(35,41)(36,57)(38,58)(39,42)(40,46)(43,52)(50,64)(53,55)(54,63)(56,60)(61,62);(1,4)(2,10)(3,16)(5,21)(6,28)(7,22)(8,23)(9,30)(11,34)(12,39)(13,35)(14,36)(15,41)(17,43)(18,48)(19,45)(20,44)(24,50)(25,55)(26,54)(27,53)(29,52)(31,56)(32,57)(33,42)(37,62)(38,61)(40,60)(46,64)(47,63)(49,51)(58,59);(1,5)(2,11)(3,17,19,47)(4,21)(6,26)(7,29)(8,27)(9,31,32,46)(10,34)(12,24)(13,40)(14,38)(15,37,33,58)(16,43,45,63)(18,25,20,49)(22,52)(23,53)(28,54)(30,56,57,64)(35,60)(36,61)(39,50)(41,62,42,59)(44,51,48,55)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319969, #72319988, #72320069
maximal normal subgroup of	#72319969, #72319988, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334504);