#149407

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,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, 56603);