#418512

order	512
nilpotency class	3
isoclinism	isoclinic class of #418512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15263
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

1630 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)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,29)(16,35)(17,38)(19,34)(20,33)(22,50)(24,41)(26,32)(27,39)(28,40)(30,37)(31,45)(36,56)(42,44)(43,60)(46,58)(47,54)(48,59)(49,53)(51,62)(52,55)(57,63)(61,64);(1,3)(2,9)(4,16)(5,26)(6,19)(7,18)(8,20)(10,35)(11,38)(12,33)(13,15)(14,34)(17,27)(21,42)(22,51)(23,48)(24,45)(25,54)(28,47)(29,46)(30,44)(31,37)(32,39)(36,61)(40,58)(41,59)(43,52)(49,60)(50,57)(53,64)(55,62)(56,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36)(12,21)(13,41)(14,37)(15,42)(17,43)(18,44)(19,48)(20,45)(25,55)(26,51)(27,52)(28,56)(29,53)(32,57)(33,59)(34,35)(38,62)(39,49)(40,50)(46,64)(47,63)(54,61)(58,60);(1,5,6,27,8,29,7,28)(2,11,12,39,14,25,13,40)(3,17,18,26,20,47,19,46)(4,22,23,52,24,53,30,56)(9,32,15,38,34,58,33,54)(10,36,21,49,37,55,41,50)(16,43,44,51,45,63,48,64)(31,57,42,62,35,60,59,61)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264934, #72264949, #72264996
maximal normal subgroup of	#72264934, #72264949, #72264996
maximal quotient of	#72265602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325708);