#481728

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1505
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1505
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

754 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)(3,14,18,43)(4,20,23,54)(5,11)(6,26,29,59)(7,31)(8,13)(9,17)(10,21,37,24)(12,25,40,30)(15,44)(16,42,19,38)(22,39)(27,33)(28,36)(32,56,62,60)(34,50)(35,51)(41,45)(46,52,49,63)(47,61)(48,64)(53,57)(55,58);(1,3)(2,9)(4,15)(5,16)(6,27)(7,19)(8,18)(10,32)(11,33)(12,38)(13,35)(14,26)(17,30)(20,52)(21,45)(22,56)(23,48)(24,47)(25,51)(28,60)(29,50)(31,34)(36,64)(37,62)(39,44)(40,42)(41,58)(43,59)(46,57)(49,55)(53,61)(54,63);(1,4)(2,10,13,37)(3,15)(5,21)(6,28)(7,24)(8,23)(9,32,35,62)(11,20,31,54)(12,39,40,36)(14,41,43,61)(16,45)(17,49)(18,48)(19,47)(22,29)(25,57)(26,58,59,53)(27,60)(30,55)(33,52,34,63)(38,44,42,64)(46,51)(50,56);(1,5,8,7)(2,11,13,31)(3,16,18,19)(4,21,23,24)(6,25,29,30)(9,33,35,34)(10,20,37,54)(12,26,40,59)(14,42,43,38)(15,45,48,47)(17,27,51,50)(22,55,28,57)(32,52,62,63)(36,53,39,58)(41,64,61,44)(46,56,49,60)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307134, #72307452, #72307468
maximal normal subgroup of	#72307134, #72307452, #72307468

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388924);