#147991

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
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

676 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)(5,6,10,17,11,15,31,41,12,16,32,42,9,28,30,55)(7,18,22,44,23,36,50,60,21,45,49,65,20,13,33,35)(8,14,26,39,25,40,53,63,27,38,54,62,29,37,56,61)(19,43,47,64,46,59,66,68,24,51,52,67,48,34,57,58);(5,7,9,20,12,21,11,23)(6,13,15,18,16,36,28,45)(8,24,25,48,27,19,29,46)(10,22,30,33,32,49,31,50)(14,34,37,51,38,59,40,43)(17,35,41,44,42,60,55,65)(26,52,53,57,54,47,56,66)(39,58,61,67,62,68,63,64);(2,4)(5,8)(6,14)(7,19)(9,29)(10,26)(11,25)(12,27)(13,34)(15,40)(16,38)(17,39)(18,43)(20,48)(21,24)(22,47)(23,46)(28,37)(30,56)(31,53)(32,54)(33,57)(35,58)(36,59)(41,63)(42,62)(44,64)(45,51)(49,52)(50,66)(55,61)(60,68)(65,67)

Polycyclic presentation:

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

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, 55187);