#479873

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17296
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,13,8,12,7,14)(3,15,19,30,18,9,20,29)(4,10,22,26,11,5,23,25)(16,27,21,17,32,31,28,24);(1,3)(2,9,12,15)(4,21,11,28)(5,24)(6,19)(7,20)(8,18)(10,17)(13,29,14,30)(16,22,32,23)(25,27)(26,31);(1,4)(2,10,12,5)(3,16)(6,23)(7,22)(8,11)(9,27,15,31)(13,25,14,26)(17,29,24,30)(18,32)(19,28)(20,21);(1,5)(2,11,12,4)(3,17,18,24)(6,26)(7,25)(8,10)(9,28)(13,22,14,23)(15,21)(16,30)(19,27,20,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387069);