#147635

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
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

1004 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,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, 54831);