#147406

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54602);