#469688

order	512
nilpotency class	3
isoclinism	isoclinic class of #469687
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

1084 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)(3,8,16,31)(4,17,20,48)(5,10,22,36)(6,24,27,11)(9,32,35,21)(13,40,18,49)(14,43,44,29)(15,38,25,30)(19,52,54,47)(23,39,26,37)(28,51,33,42)(34,55,60,53)(41,61,50,64)(45,57,46,58)(56,62,59,63);(1,3)(2,8)(4,18)(5,14)(6,25)(7,16)(9,33)(10,29)(11,38)(12,31)(13,20)(15,27)(17,40)(19,50)(21,42)(22,44)(23,46)(24,30)(26,45)(28,35)(32,51)(34,59)(36,43)(37,58)(39,57)(41,54)(47,61)(48,49)(52,64)(53,62)(55,63)(56,60);(1,4,6,21,7,20,27,32)(2,9,11,17,12,35,24,48)(3,13,15,42,16,18,25,51)(5,19,23,53,22,54,26,55)(8,28,30,40,31,33,38,49)(10,34,37,52,36,60,39,47)(14,41,45,62,44,50,46,63)(29,56,57,64,43,59,58,61);(1,5)(2,10)(3,14)(4,19,20,54)(6,26)(7,22)(8,29)(9,34,35,60)(11,39)(12,36)(13,41,18,50)(15,46)(16,44)(17,47,48,52)(21,55,32,53)(23,27)(24,37)(25,45)(28,56,33,59)(30,58)(31,43)(38,57)(40,61,49,64)(42,63,51,62)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286082, #72286090, #72286267
maximal normal subgroup of	#72286082, #72286090, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376884);