#394601

order	512
nilpotency class	3
isoclinism	isoclinic class of #394600
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2078 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)(5,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,30)(17,41)(19,33)(20,32)(22,49)(23,51)(24,36)(26,35)(27,39)(28,38)(29,37)(31,45)(34,52)(40,44)(42,61)(43,56)(46,58)(47,57)(48,53)(50,55)(54,59)(60,62)(63,64);(1,3,8,20)(2,9,14,33)(4,22,26,16)(5,17,28,46)(6,19,7,18)(10,30,36,49)(11,31,38,57)(12,32,13,15)(21,40,35,56)(23,50,53,42)(24,43,25,44)(27,47,29,45)(34,55,59,61)(37,58,39,41)(48,60,51,64)(52,62,54,63);(1,4,6,24,8,26,7,25)(2,10,12,35,14,36,13,21)(3,16,18,43,20,22,19,44)(5,23,27,52,28,53,29,54)(9,30,15,40,33,49,32,56)(11,34,37,51,38,59,39,48)(17,42,45,62,46,50,47,63)(31,55,41,60,57,61,58,64);(1,5)(2,11)(3,17)(4,23)(6,27)(7,29)(8,28)(9,31)(10,34)(12,37)(13,39)(14,38)(15,41)(16,42)(18,45)(19,47)(20,46)(21,48)(22,50)(24,52)(25,54)(26,53)(30,55)(32,58)(33,57)(35,51)(36,59)(40,60)(43,62)(44,63)(49,61)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317787, #72318063, #72318571
maximal normal subgroup of	#72317787, #72318063, #72318571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301797);