#147949

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2011
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11022
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

953 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,f^b=fh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55145);