#423224

order	512
nilpotency class	3
isoclinism	isoclinic class of #423223
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15313
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60645	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264941, #72265041
maximal normal subgroup of	#72264941, #72265041
maximal quotient of	#72314066

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330420);