#469686

order	512
nilpotency class	3
isoclinism	isoclinic class of #469685
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16660
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

1087 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)(3,8,15,31)(4,9,19,35)(5,21,22,40)(6,24,28,11)(10,27,37,23)(13,33,17,29)(14,43,44,53)(16,38,25,32)(18,49,50,54)(20,39,26,36)(30,47,57,45)(34,52,60,51)(41,61,48,64)(42,56,46,58)(55,62,59,63);(1,3)(2,8)(4,17)(5,14)(6,25)(7,15)(9,33)(10,30)(11,38)(12,31)(13,19)(16,28)(18,48)(20,42)(21,53)(22,44)(23,47)(24,32)(26,46)(27,45)(29,35)(34,59)(36,56)(37,57)(39,58)(40,43)(41,50)(49,64)(51,62)(52,63)(54,61)(55,60);(1,4)(2,9)(3,13,15,17)(5,18,22,50)(6,26)(7,19)(8,29,31,33)(10,34,37,60)(11,39)(12,35)(14,41)(16,46,25,42)(20,28)(21,54,40,49)(23,52,27,51)(24,36)(30,55)(32,58,38,56)(43,61)(44,48)(45,63)(47,62)(53,64)(57,59);(1,5)(2,10)(3,14)(4,18,19,50)(6,27)(7,22)(8,30)(9,34,35,60)(11,40)(12,37)(13,41,17,48)(15,44)(16,47)(20,52,26,51)(21,24)(23,28)(25,45)(29,55,33,59)(31,57)(32,53)(36,54,39,49)(38,43)(42,63,46,62)(56,61,58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285951, #72286066, #72286090
maximal normal subgroup of	#72285951, #72286066, #72286090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376882);