#424605

order	512
nilpotency class	3
isoclinism	isoclinic class of #424604
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15323
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 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)(3,13,16,8)(4,9)(5,22)(6,25)(7,11)(10,23)(12,28)(14,29,18,39)(15,42,43,54)(17,31,26,37)(19,48)(20,33)(21,52)(24,38)(27,36)(30,44,35,45)(32,50)(34,51)(40,61,46,64)(41,59,47,56)(49,60)(53,58)(55,63,57,62);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,29)(10,35)(11,13)(12,37)(14,20)(17,28)(19,46)(21,41)(22,54)(23,45)(24,43)(25,31)(27,44)(30,36)(32,57)(33,39)(34,59)(38,42)(40,49)(47,51)(48,61)(50,62)(52,56)(53,63)(55,58)(60,64);(1,4)(2,9)(3,14,16,18)(5,19)(6,21)(7,20)(8,29,13,39)(10,32)(11,33)(12,34)(15,40,43,46)(17,41,26,47)(22,48)(23,50)(24,49)(25,52)(27,53)(28,51)(30,55,35,57)(31,56,37,59)(36,58)(38,60)(42,64,54,61)(44,62,45,63);(1,5,7,24)(2,10,11,36)(3,15,16,43)(4,19,20,49)(6,27,28,23)(8,30,13,35)(9,32,33,58)(12,38,25,22)(14,40,18,46)(17,45,26,44)(21,53,51,50)(29,55,39,57)(31,42,37,54)(34,60,52,48)(41,63,47,62)(56,64,59,61)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^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,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331801);