#147675

order	512
nilpotency class	4
isoclinism	isoclinic class of #136643
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
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

994 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)(5,6,10,15,11,17,33,16)(7,18,21,43,22,48,32,47)(8,13,26,41,27,42,25,40)(9,28,30,14,31,44,61,45)(12,29,35,20,37,52,36,53)(19,46,50,66,51,67,23,54)(24,56,59,39,60,64,58,65)(34,55,62,49,63,68,38,57);(5,7,11,22)(6,12,17,37)(8,23,27,50)(9,29,31,52)(10,32,33,21)(13,38,42,62)(14,43,45,47)(15,36,16,35)(18,28,48,44)(19,26,51,25)(20,30,53,61)(24,57,60,49)(34,41,63,40)(39,46,65,67)(54,56,66,64)(55,59,68,58);(2,4)(5,8)(6,13)(7,19)(9,24)(10,26)(11,27)(12,34)(14,39)(15,41)(16,40)(17,42)(18,46)(20,49)(21,50)(22,51)(23,32)(25,33)(28,56)(29,55)(30,59)(31,60)(35,62)(36,38)(37,63)(43,66)(44,64)(45,65)(47,54)(48,67)(52,68)(53,57)(58,61)

Polycyclic presentation:

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

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, 54871);