#493027

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×C₄×#99
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281046, #72283959, #72312316, …
maximal normal subgroup of	#72281046, #72283959, #72312316, …
maximal quotient of	C₂×#493027

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400223);