#147537

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,17,11,15,33,44,12,16,34,45,9,29,31,58)(7,18,22,48,20,13,35,37,21,38,52,63,30,39,59,64)(8,23,27,46,26,42,57,67,28,14,32,43,25,41,56,66)(19,47,51,68,49,36,60,61,50,40,53,65,24,54,55,62);(2,4)(5,7)(6,13)(8,24)(9,30)(10,22)(11,20)(12,21)(14,40)(15,38)(16,39)(17,37)(18,29)(19,26)(23,47)(25,50)(27,55)(28,49)(31,59)(32,60)(33,35)(34,52)(36,42)(41,54)(43,65)(44,63)(45,64)(46,68)(48,58)(51,57)(53,56)(61,67)(62,66);(5,8)(6,14)(7,19)(9,25)(10,32)(11,26)(12,28)(13,36)(15,41)(16,23)(17,46)(18,47)(20,49)(21,50)(22,53)(24,30)(27,34)(29,42)(31,57)(33,56)(35,55)(37,62)(38,40)(39,54)(43,45)(44,67)(48,65)(51,52)(58,66)(59,60)(61,64)(63,68)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54733);