#494416

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1869
prime radical	#3449	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239242, #72249825, #72249827, …
maximal normal subgroup of	#72239242, #72249825, #72249827, …
maximal quotient of	#72239259, #72322881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401612);