#432901

order	512
nilpotency class	3
isoclinism	isoclinic class of #432901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15468
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

962 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)(3,8,15,31)(4,9,18,34)(5,20)(6,24)(7,11)(10,23)(12,28)(13,29)(14,43,21,52)(16,37,25,32)(17,47,49,53)(19,38,26,35)(22,39)(27,36)(30,46,57,44)(33,51,59,50)(40,60)(41,56)(42,58)(45,55)(48,63)(54,61)(62,64);(1,3)(2,8,11,31)(4,13,18,41)(5,21,22,14)(6,25)(7,15)(9,29)(10,30)(12,37,24,32)(16,28)(17,48)(19,45,26,42)(20,52)(23,44,27,46)(33,54,59,64)(34,56)(35,58)(36,57)(38,55)(39,43)(40,49)(47,63,53,60)(50,61)(51,62);(1,4)(2,9,11,34)(3,13,15,41)(5,17)(6,26)(7,18)(8,29)(10,33,36,59)(12,38,24,35)(14,40,21,48)(16,45,25,42)(19,28)(20,53,39,47)(22,49)(23,51)(27,50)(30,54)(31,56)(32,58)(37,55)(43,63)(44,62,46,61)(52,60)(57,64);(1,5)(2,10)(3,14)(4,17)(6,27)(7,22)(8,30)(9,33)(11,36)(12,39)(13,40)(15,21)(16,46)(18,49)(19,51)(20,24)(23,28)(25,44)(26,50)(29,54)(31,57)(32,52)(34,59)(35,53)(37,43)(38,47)(41,48)(42,62)(45,61)(55,63)(56,64)(58,60)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319622, #72319962, #72320088
maximal normal subgroup of	#72319622, #72319962, #72320088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340097);