#473982

order	512
nilpotency class	3
isoclinism	isoclinic class of #467073
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2640
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16891
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 381178);