#476488

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17090
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383684);