#455955

order	512
nilpotency class	3
isoclinism	isoclinic class of #455954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16272
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1115 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)(3,14)(4,10)(5,11)(6,27)(7,13)(8,12)(9,17)(15,42)(16,47)(18,44)(19,32)(20,30)(21,34)(22,36)(23,37)(24,56)(25,40)(26,41)(28,39)(29,35)(31,49)(33,46)(38,55)(43,50)(45,59)(48,57)(51,60)(52,58)(53,62)(54,61)(63,64);(1,3,8,19,7,18,6,17)(2,9,27,44,13,32,12,14)(4,20,22,46,23,52,29,15)(5,16,25,49,26,50,28,51)(10,33,35,30,37,42,36,58)(11,31,39,47,41,60,40,43)(21,53,55,64,54,48,24,45)(34,59,56,57,61,63,38,62);(1,4,6,22,7,23,8,29)(2,10,12,35,13,37,27,36)(3,15,17,20,18,46,19,52)(5,24,25,54,26,55,28,21)(9,30,14,42,32,58,44,33)(11,38,39,61,41,56,40,34)(16,48,49,64,50,53,51,45)(31,59,47,62,60,63,43,57);(1,5,7,26)(2,11,13,41)(3,16,18,50)(4,21,23,54)(6,28,8,25)(9,31,32,60)(10,34,37,61)(12,40,27,39)(14,43,44,47)(15,45,46,64)(17,51,19,49)(20,53,52,48)(22,55,29,24)(30,57,58,62)(33,59,42,63)(35,56,36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287261, #72287312, #72287345
maximal normal subgroup of	#72287261, #72287312, #72287345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363151);