#108167

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15363);