#146806

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×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

958 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:

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

Polycyclic presentation:

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

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, 54002);