#409171

order	512
nilpotency class	3
isoclinism	isoclinic class of #400512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15009
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,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, 316367);