#506210

order	512
nilpotency class	4
isoclinism	isoclinic class of #500851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28097
prime radical	#24287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 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)(3,9)(4,20)(5,26)(6,12)(7,11)(8,13)(10,23)(14,31)(15,43)(16,19)(17,35)(18,34)(21,53)(22,39)(24,40)(25,37)(27,33)(28,47)(29,60)(30,41)(32,46)(36,50)(38,54)(42,61)(44,49)(45,62)(48,51)(52,56)(55,58)(57,64)(59,63);(1,3,14,42,7,17,30,51,8,19,28,59,5,27,6,18)(2,9,31,61,11,35,41,48,13,16,47,63,26,33,12,34)(4,21,54,45,25,50,24,49,22,56,29,15,23,58,57,46)(10,36,64,44,20,52,38,43,37,55,40,32,39,53,60,62);(1,4)(2,10)(3,15)(5,25)(6,29)(7,23)(8,22)(9,32)(11,39)(12,40)(13,37)(14,24)(16,44)(17,49)(18,50)(19,45)(20,26)(21,42)(27,46)(28,57)(30,54)(31,38)(33,62)(34,52)(35,43)(36,61)(41,64)(47,60)(48,53)(51,58)(55,63)(56,59);(1,2)(3,16)(4,10)(5,26)(6,12)(7,11)(8,13)(9,19)(14,31)(15,44)(17,33)(18,48)(20,25)(21,55)(22,37)(23,39)(24,38)(27,35)(28,47)(29,40)(30,41)(32,45)(34,51)(36,56)(42,63)(43,46)(49,62)(50,53)(52,58)(54,64)(57,60)(59,61)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, C₂×#506210, #72342994, …
maximal normal subgroup of	#72217287, C₂×#506210, #72342994, …
maximal quotient of	C₂×#506210

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413406);