#146541

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

970 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53737);