#148737

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2085
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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)(7,18,22,37)(8,23)(9,14)(10,31)(11,17)(12,20,36,52)(13,26)(15,29)(16,30)(19,47,24,57)(21,35,54,66)(25,42)(27,61)(28,56)(32,44)(33,41)(34,64,38,53)(39,59)(40,60)(43,62)(45,58,67,49)(46,68,63,51)(48,55,50,65);(1,2)(3,4)(5,7,11,22)(6,12)(8,24)(9,20,30,52)(10,21,33,54)(13,38,40,34)(14,18)(15,35)(16,37)(17,36)(19,28)(23,55,56,65)(25,50)(26,58)(27,48)(29,51,62,68)(31,63)(32,64)(39,67,44,45)(41,46)(42,57,61,47)(43,66)(49,60)(53,59);(2,4)(5,8,9,25,11,28,30,27)(6,13,14,39,17,40,16,44)(7,19,20,48,22,24,52,50)(10,32,29,26,33,59,62,60)(12,34,18,45,36,38,37,67)(15,42,41,56,43,61,31,23)(21,53,51,49,54,64,68,58)(35,47,46,55,66,57,63,65)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=eh,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, 55933);