#148648

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55844);