#493794

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1804
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1025 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)(3,14,18,33)(4,19)(5,10)(6,26)(7,13)(8,16,32,49)(9,22)(11,24)(12,25)(15,43,47,64)(17,31,27,38)(20,39)(21,52)(23,53)(28,35)(29,37)(30,59,62,50)(34,54)(36,56)(40,58)(41,55,60,46)(42,57,63,48)(44,51,45,61);(1,3,7,18)(2,8)(4,15,23,47)(5,16,25,49)(6,27,29,17)(9,30)(10,14)(11,38)(12,33)(13,32)(19,51)(20,44,21,45)(22,55,56,46)(24,57,58,48)(26,42)(28,59,54,50)(31,40)(34,41)(35,60)(36,62)(37,63)(39,64)(43,52)(53,61);(1,4)(2,9)(3,15)(5,20)(6,28)(7,23)(8,30)(10,34)(11,39)(12,35)(13,36)(14,41)(16,44)(17,50)(18,47)(19,26)(21,25)(22,24)(27,59)(29,54)(31,43)(32,62)(33,60)(37,53)(38,64)(40,52)(42,51)(45,49)(46,48)(55,57)(56,58)(61,63);(1,5,7,25)(2,10,13,12)(3,16,18,49)(4,20,23,21)(6,24,29,58)(8,14,32,33)(9,34,36,35)(11,37,40,26)(15,44,47,45)(17,48,27,57)(19,39,53,52)(22,54,56,28)(30,41,62,60)(31,42,38,63)(43,51,64,61)(46,59,55,50)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307107, #72307113, #72307124, …
maximal normal subgroup of	#72307107, #72307113, #72307124, …
maximal quotient of	C₂×#493794

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400990);