#147613

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10966
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54809);