#464655

order	512
nilpotency class	3
isoclinism	isoclinic class of #464654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16508
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1453 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^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,h^c=hi,e^d=ei

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264949, #72264954, #72265047
maximal normal subgroup of	#72264949, #72264954, #72265047
maximal quotient of	#72314124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371851);