#404531

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14893
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 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)(3,9)(4,19,21,36)(5,24,25,38)(6,12)(7,14)(8,13)(10,22,35,23)(11,26,37,27)(15,40,42,49)(16,44,45,54)(17,31)(18,32)(20,34)(28,39)(29,43,33,48)(30,47,56,46)(41,57)(50,53)(51,58)(52,59)(55,63)(60,61)(62,64);(1,3,6,17,8,18,7,28)(2,9,12,31,13,32,14,39)(4,15,23,48,21,42,22,43)(5,16,27,47,25,45,26,46)(10,33,36,40,35,29,19,49)(11,30,38,44,37,56,24,54)(20,41,51,61,52,63,53,62)(34,57,58,60,59,55,50,64);(1,4)(2,10)(3,15)(5,20)(6,23)(7,22)(8,21)(9,29)(11,34)(12,36)(13,35)(14,19)(16,41)(17,48)(18,42)(24,50)(25,52)(26,53)(27,51)(28,43)(30,55)(31,49)(32,33)(37,59)(38,58)(39,40)(44,64)(45,63)(46,62)(47,61)(54,60)(56,57);(1,5)(2,11,13,37)(3,16)(4,20)(6,27)(7,26)(8,25)(9,30,32,56)(10,34,35,59)(12,38,14,24)(15,41)(17,47)(18,45)(19,50,36,58)(21,52)(22,53)(23,51)(28,46)(29,55,33,57)(31,44,39,54)(40,60,49,64)(42,63)(43,62)(48,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285784, #72285996, …
maximal normal subgroup of	#72285525, #72285784, #72285996, …
maximal quotient of	C₂×#404531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311727);