#405383

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14912
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1218 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)(3,13)(4,18)(5,10)(6,24)(7,12)(8,16)(9,21)(11,26)(14,36)(15,37)(17,35)(19,31)(20,32)(22,49)(23,34)(25,29)(27,39)(28,41)(30,47)(33,51)(38,59)(40,55)(42,57)(43,46)(44,48)(45,62)(50,53)(52,63)(54,60)(56,64)(58,61);(1,3)(2,8)(4,14,21,40)(5,15)(6,25)(7,17)(9,30,18,27)(10,28)(11,35)(12,29)(13,24)(16,26)(19,38,46,61)(20,47,48,39)(22,50)(23,42)(31,54,43,52)(32,36,44,55)(33,57)(34,53)(37,49)(41,51)(45,63,64,60)(56,58,62,59);(1,4,6,20,7,21,26,48)(2,9,11,32,12,18,24,44)(3,14,16,39,17,40,25,47)(5,19,22,45,23,46,51,64)(8,27,13,36,29,30,35,55)(10,31,33,56,34,43,49,62)(15,38,41,60,42,61,50,63)(28,52,37,58,53,54,57,59);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,28)(9,31)(11,33)(12,34)(13,37)(14,38)(16,41)(17,42)(18,43)(20,45)(21,46)(24,49)(25,50)(26,51)(27,52)(29,53)(30,54)(32,56)(35,57)(36,58)(39,60)(40,61)(44,62)(47,63)(48,64)(55,59)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312579);