#421763

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15297
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1479 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328959);