#452540

order	512
nilpotency class	3
isoclinism	isoclinic class of #452539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16051
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

870 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318430, #72318935, #72318981
maximal normal subgroup of	#72318430, #72318935, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359736);