#148897

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2054
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(7,19)(8,25)(9,16)(10,15)(11,34)(12,17)(13,21)(14,27)(18,30)(20,39)(22,38)(23,36)(24,48)(26,42)(28,40)(29,56)(31,45)(32,47)(33,43)(35,52)(37,54)(41,61)(44,53)(46,50)(49,55)(51,63)(57,58)(59,62)(60,64)(65,68)(66,67);(5,7,10,32,11,23,31,54,12,24,33,22,30,53,9,21)(6,13,16,44,18,38,43,48,17,37,45,36,34,47,15,19)(8,20,27,60,28,51,59,68,29,52,61,50,58,67,26,49)(14,39,25,55,42,66,57,46,41,35,56,65,62,63,40,64);(5,8,9,26,30,58,33,61,12,29,31,59,11,28,10,27)(6,14,15,40,34,62,45,56,17,41,43,57,18,42,16,25)(7,20,21,49,53,67,22,50,24,52,54,68,23,51,32,60)(13,35,19,46,47,66,36,55,37,39,48,64,38,63,44,65)

Polycyclic presentation:

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

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, 56093);