#148288

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
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

658 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:

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

Polycyclic presentation:

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

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, 55484);