#147234

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10885
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,e^c=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, 54430);