#411070

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15103
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 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)(4,17)(5,11)(6,9)(7,12)(10,19)(13,22)(14,28)(15,23)(16,24)(18,26)(20,25)(21,30)(27,32)(29,31);(1,3,5,16)(2,8)(4,14,20,29)(6,15,7,13)(9,23)(10,21)(11,24)(12,22)(17,31)(18,30,19,27)(25,28)(26,32);(2,9,11,12)(3,13,16,15)(10,17,26,25)(14,27,29,30);(1,4,6,18,5,20,7,19)(2,10,12,25,11,26,9,17)(3,14,15,30,16,29,13,27)(8,21,22,28,24,32,23,31)

Polycyclic presentation:

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

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, 318266);