#473920

order	512
nilpotency class	3
isoclinism	isoclinic class of #467073
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2640
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16887
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,d^2=e,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,f^b=fi,d^c=de,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381116);