#404128

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	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1303 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)(5,26)(6,14)(7,12)(8,13)(9,18)(10,24)(11,28)(16,34)(17,31)(19,33)(20,32)(22,35)(23,37)(25,36)(27,39)(29,38)(30,44)(40,43)(41,46)(42,61)(45,53)(47,55)(48,56)(49,50)(51,59)(52,58)(54,63)(57,62)(60,64);(1,3,6,18,8,20,7,19)(2,9,12,15,13,33,14,32)(4,16,23,43,25,44,24,48)(5,17,27,45,29,47,28,46)(10,34,36,56,37,30,21,40)(11,31,38,41,39,55,26,53)(22,42,50,62,51,64,52,63)(35,57,58,61,59,54,49,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30)(11,35)(12,36)(13,37)(14,21)(15,40)(17,42)(18,43)(19,48)(20,44)(26,49)(27,50)(28,52)(29,51)(31,54)(32,56)(33,34)(38,58)(39,59)(41,60)(45,62)(46,63)(47,64)(53,61)(55,57);(1,5)(2,11,13,39)(3,17)(4,22)(6,27)(7,28)(8,29)(9,31,33,55)(10,35,37,59)(12,38,14,26)(15,41,32,53)(16,42)(18,45)(19,46)(20,47)(21,49,36,58)(23,50)(24,52)(25,51)(30,54,34,57)(40,60,56,61)(43,62)(44,64)(48,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285753, #72285996, …
maximal normal subgroup of	#72285525, #72285753, #72285996, …
maximal quotient of	C₂×#404128

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311324);