#404108

order	512
nilpotency class	3
isoclinism	isoclinic class of #395191
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14886
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1315 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)(4,10)(5,24)(6,14)(7,12)(8,13)(9,18)(11,27)(16,42)(17,30)(19,32)(20,31)(21,50)(22,39)(23,36)(25,38)(26,37)(28,35)(29,49)(33,44)(34,51)(40,45)(41,47)(43,56)(46,53)(48,57)(52,54)(55,59)(58,63)(60,64)(61,62);(1,3,6,18,8,19,7,20)(2,9,12,15,13,31,14,32)(4,16,22,44,23,45,28,49)(5,17,25,46,26,48,27,47)(10,33,35,42,36,29,39,40)(11,30,37,41,38,57,24,53)(21,43,51,62,52,63,55,64)(34,58,50,61,59,56,54,60);(1,4)(2,10)(3,16)(5,21)(6,22)(7,28)(8,23)(9,29)(11,34)(12,35)(13,36)(14,39)(15,40)(17,43)(18,44)(19,45)(20,49)(24,54)(25,51)(26,52)(27,55)(30,56)(31,33)(32,42)(37,50)(38,59)(41,60)(46,62)(47,64)(48,63)(53,61)(57,58);(1,5)(2,11,13,38)(3,17)(4,21)(6,25)(7,27)(8,26)(9,30,31,57)(10,34,36,59)(12,37,14,24)(15,41,32,53)(16,43)(18,46)(19,48)(20,47)(22,51)(23,52)(28,55)(29,56,33,58)(35,50,39,54)(40,60,42,61)(44,62)(45,63)(49,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285582, #72286119, …
maximal normal subgroup of	#72285525, #72285582, #72286119, …
maximal quotient of	C₂×#404108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311304);