#147970

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=ei,d^c=di,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, 55166);