#431173

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15432
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

811 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338369);