#423438

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15314
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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)(3,8)(4,9,18,31)(5,20)(6,23)(7,11)(10,22)(12,25)(13,35,37,26)(14,39)(15,28)(16,43)(17,45,47,58)(19,50,49,32)(21,34)(24,33)(27,41)(29,42)(30,51,57,48)(36,60,46,64)(38,53,62,59)(40,55)(44,54)(52,61,56,63);(1,3)(2,8)(4,13,18,37)(5,14)(6,16)(7,15)(9,26,31,35)(10,27)(11,28)(12,29)(17,46,47,36)(19,38,49,62)(20,39)(21,40)(22,41)(23,43)(24,44)(25,42)(30,56,57,52)(32,53,50,59)(33,54)(34,55)(45,60,58,64)(48,63,51,61);(1,4)(2,9,11,31)(3,13,15,37)(5,17)(6,19)(7,18)(8,26)(10,30,33,57)(12,32,23,50)(14,36,40,46)(16,38,42,62)(20,45,34,58)(21,47)(22,48)(24,51)(25,49)(27,52)(28,35)(29,53)(39,64)(41,61,44,63)(43,59)(54,56)(55,60);(1,5)(2,10)(3,14)(4,17)(6,24)(7,21)(8,27)(9,30)(11,33)(12,34)(13,36)(15,40)(16,44)(18,47)(19,51)(20,23)(22,25)(26,52)(28,54)(29,55)(31,57)(32,58)(35,56)(37,46)(38,63)(39,43)(41,42)(45,50)(48,49)(53,60)(59,64)(61,62)

Polycyclic presentation:

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

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, 330634);