#479764

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17238
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

624 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=i,f^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,g^b=gi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

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, 386960);