#147703

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

996 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)(5,6,10,15,11,17,34,16)(7,18,21,43,22,48,33,47)(8,23,26,42,28,41,27,13)(9,29,31,14,32,44,64,45)(12,30,36,20,38,53,37,54)(19,49,24,57,52,66,51,46)(25,59,61,67,63,40,62,56)(35,60,65,68,55,50,39,58);(5,7,11,22)(6,12,17,38)(8,24,28,51)(9,30,32,53)(10,33,34,21)(13,39,42,65)(14,43,45,47)(15,37,16,36)(18,29,48,44)(19,27,52,26)(20,31,54,64)(23,55,41,35)(25,60,63,50)(40,57,59,46)(49,67,66,56)(58,62,68,61);(2,4)(5,8)(6,13)(7,19)(9,25)(10,27)(11,28)(12,35)(14,40)(15,41)(16,23)(17,42)(18,46)(20,50)(21,51)(22,52)(24,33)(26,34)(29,56)(30,58)(31,62)(32,63)(36,39)(37,65)(38,55)(43,66)(44,67)(45,59)(47,49)(48,57)(53,68)(54,60)(61,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54899);