#411399

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15115
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

722 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)(3,14)(4,19)(5,18)(6,12)(7,11)(8,13)(9,16)(10,23)(15,30)(17,25)(20,21)(22,29)(24,28)(26,32)(27,31);(1,3,8,18)(2,9,13,25)(4,20)(5,12,14,11)(6,17,7,16)(10,27)(15,23)(19,30)(21,29)(22,32)(24,31)(26,28);(1,4,6,22,8,24,7,23)(2,10,11,28,13,29,12,19)(3,15,16,31,18,32,17,20)(5,21,9,26,14,27,25,30);(1,5,6,9,8,14,7,25)(2,3,12,17,13,18,11,16)(4,21,22,26,24,27,23,30)(10,15,19,20,29,32,28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318595);