#148946

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2085
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

436 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fh,g^b=gi,h^b=hi,f^c=fh,g^c=gi,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, 56142);