#403041

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14865
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1208 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:

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

Polycyclic presentation:

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

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, 310237);