#149408

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

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,12)(8,10)(9,11)(13,14,17,24,18,25,33,22,19,26,34,21,32,40,16,23)(15,27,28,36,30,41,43,42,31,37,44,20,35,38,29,39);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,31)(14,20,26,36)(16,28,33,44)(17,29,34,43)(18,35,32,30)(21,27,24,37)(22,41,23,38)(25,42,40,39);(2,4)(5,8,9,7)(6,10,11,12)(14,21)(15,28)(16,17)(18,32)(20,36)(22,25)(23,40)(24,26)(27,38)(29,30)(31,44)(33,34)(35,43)(37,41)

Polycyclic presentation:

2^9|a^2=g,b^2=dehi,c^2=deh,d^2=hi,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=ghi,h^c=hi

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, 56604);