#149104

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

430 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56300);