#392722

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14694
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3610 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)(3,15,19,32)(4,21,24,10)(5,25,28,36)(6,12)(7,14)(8,13)(9,18,33,20)(11,27,37,26)(16,39)(17,44)(22,51)(23,38,29,35)(30,42)(31,46)(34,52)(40,45)(41,60,62,57)(43,58)(47,55)(48,49)(50,54)(53,56)(59,61,64,63);(1,3,6,20,8,19,7,18)(2,9,12,32,13,33,14,15)(4,16,29,42,24,43,23,48)(5,17,27,45,28,46,26,47)(10,30,38,58,21,49,35,39)(11,31,25,55,37,44,36,40)(22,41,52,63,53,62,54,61)(34,57,56,64,50,60,51,59);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,34)(12,38)(13,21)(14,35)(15,39)(17,41)(18,48)(19,43)(20,42)(25,56)(26,54)(27,52)(28,53)(31,57)(32,58)(33,49)(36,51)(37,50)(40,59)(44,60)(45,63)(46,62)(47,61)(55,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,26)(8,28)(9,31)(10,34)(12,25)(13,37)(14,36)(15,40)(16,41)(18,47)(19,46)(20,45)(21,50)(23,54)(24,53)(29,52)(30,57)(32,55)(33,44)(35,51)(38,56)(39,59)(42,63)(43,62)(48,61)(49,60)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342961, #72342967, #72342979
maximal normal subgroup of	#72342961, #72342967, #72342979
maximal quotient of	#72343033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299918);