#398814

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14769
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 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:

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

Polycyclic presentation:

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

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, 306010);