#500008

order	512
nilpotency class	4
isoclinism	isoclinic class of #500007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23960
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24286	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

890 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,e^c=ei,f^c=fh,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250025, #72250066, #72281018, …
maximal normal subgroup of	#72250025, #72250066, #72281018, …
maximal quotient of	#72250249, #72280589, #72282210

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407204);