#479778

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17241
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

623 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)(3,8,14,27)(4,16)(5,10,17,29)(6,21)(7,11)(9,13)(12,24)(15,32,19,26)(18,31)(20,25,23,30)(22,28);(1,3,7,14)(2,8)(4,17,18,5)(6,15,24,19)(9,25)(10,16)(11,27)(12,26)(13,23,22,20)(21,32)(28,30)(29,31);(1,4,6,22,7,18,24,13)(2,9,21,16,11,28,12,31)(3,5,19,20,14,17,15,23)(8,25,26,29,27,30,32,10);(1,5)(2,10)(3,13)(4,15)(6,23)(7,17)(8,9)(11,29)(12,25)(14,22)(16,32)(18,19)(20,24)(21,30)(26,31)(27,28)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,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, 386974);