#400697

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14806
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1388 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)(5,12)(6,11)(7,13)(8,16)(9,17)(10,21)(15,25)(18,24)(20,27)(22,26)(23,30)(28,29)(31,32);(1,3,5,18,7,16,6,17)(2,8,11,24,13,14,12,9)(4,15,22,29,20,31,21,30)(10,23,27,32,26,28,19,25);(1,3)(2,9)(4,15)(5,18)(6,17)(7,16)(8,11)(10,25)(12,14)(13,24)(19,28)(20,31)(21,30)(22,29)(23,27)(26,32);(1,4,5,22,7,20,6,21)(2,10,11,27,13,26,12,19)(3,15,18,29,16,31,17,30)(8,23,24,32,14,28,9,25)

Polycyclic presentation:

2^9|b^2=ef,d^2=ef,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,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 307893);