#148902

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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)(7,18)(8,24,27,40)(9,29)(10,32)(11,17)(12,21)(13,26,37,25)(14,19)(15,39)(16,30)(20,47,50,61)(22,54)(23,34)(28,59,57,56)(31,41)(33,51,60,67)(35,63)(36,53)(38,46,55,58)(42,49,62,48)(43,45)(44,52)(64,65,66,68);(1,2)(3,4)(5,7,19,45,10,22,31,53,11,23,39,63,30,52,9,21)(6,12,29,44,16,35,15,34,17,36,41,54,32,43,14,18)(8,20,46,65,28,51,26,49,27,50,58,68,57,67,25,48)(13,33,56,64,38,61,40,62,37,60,59,66,55,47,24,42);(2,4)(5,8,9,25,30,57,39,58,11,27,31,26,10,28,19,46)(6,13,14,24,32,55,41,59,17,37,15,40,16,38,29,56)(7,20,21,48,52,67,63,68,23,50,53,49,22,51,45,65)(12,33,18,42,43,47,54,66,36,60,34,62,35,61,44,64)

Polycyclic presentation:

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

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, 56098);