#148617

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1893
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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,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, 55813);