#394629

order	512
nilpotency class	3
isoclinism	isoclinic class of #394628
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

2084 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)(3,14)(4,20)(5,11)(6,29)(7,12)(8,13)(9,17)(10,24)(15,40)(16,39)(18,32)(19,38)(21,30)(22,34)(23,35)(25,48)(26,53)(27,37)(28,36)(31,44)(33,43)(41,57)(42,55)(45,59)(46,56)(47,52)(49,64)(50,63)(51,58)(54,61)(60,62);(1,3,8,19)(2,9,13,32)(4,21,24,42)(5,16,27,45)(6,18,7,17)(10,33,20,40)(11,31,37,56)(12,38,29,14)(15,23,43,25)(22,49,52,61)(26,46,28,44)(30,48,55,35)(34,57,47,60)(36,59,53,39)(41,50,62,51)(54,63,64,58);(1,4,6,23,8,24,7,25)(2,10,12,35,13,20,29,48)(3,15,17,42,19,43,18,21)(5,22,26,50,27,52,28,51)(9,30,14,33,32,55,38,40)(11,34,36,58,37,47,53,63)(16,41,44,61,45,62,46,49)(31,54,39,57,56,64,59,60);(1,5)(2,11)(3,16)(4,22)(6,26)(7,28)(8,27)(9,31)(10,34)(12,36)(13,37)(14,39)(15,41)(17,44)(18,46)(19,45)(20,47)(21,49)(23,50)(24,52)(25,51)(29,53)(30,54)(32,56)(33,57)(35,58)(38,59)(40,60)(42,61)(43,62)(48,63)(55,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306450, #72306453, #72306472
maximal normal subgroup of	#72306450, #72306453, #72306472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301825);