#382442

order	512
nilpotency class	3
isoclinism	isoclinic class of #382441
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14335
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3369 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)(3,9)(4,19)(5,11)(6,13)(7,12)(8,14)(10,23)(15,34)(16,31)(17,42)(18,33)(20,30)(21,49)(22,37)(24,36)(25,54)(26,40)(27,39)(28,41)(29,32)(35,52)(38,53)(43,61)(44,62)(45,63)(46,64)(47,60)(48,51)(50,57)(55,58)(56,59);(1,3)(2,9)(4,20)(5,16)(6,17)(7,29)(8,18)(10,34)(11,31)(12,32)(13,42)(14,33)(15,24)(19,48)(21,50)(22,51)(23,44)(25,55)(26,46)(27,56)(28,47)(30,37)(35,61)(36,62)(38,63)(39,59)(40,64)(41,60)(43,53)(45,52)(49,58)(54,57);(1,4)(2,10,14,36)(3,15,18,44)(5,25,28,21)(6,23)(7,24)(8,22)(9,30)(11,38)(12,19,13,37)(16,45)(17,20,29,51)(26,53,27,52)(31,58,60,57)(32,34)(33,48)(35,41)(39,54)(40,49)(42,62)(43,47)(46,55)(50,56)(59,63,64,61);(1,5,8,28)(2,11,14,41)(3,16,18,47)(4,21,22,25)(6,26,7,27)(9,31,33,60)(10,35,36,38)(12,39,13,40)(15,43,44,45)(17,46,29,56)(19,49,37,54)(20,50,51,55)(23,52,24,53)(30,57,48,58)(32,59,42,64)(34,61,62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310794, #72310890, #72310893
maximal normal subgroup of	#72310794, #72310890, #72310893

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289638);