#468136

order	512
nilpotency class	3
isoclinism	isoclinic class of #467073
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2640
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16602
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 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)(3,13)(4,17)(5,8)(6,11)(7,12)(9,16)(10,21)(14,30)(15,23)(18,29)(19,28)(20,27)(22,24)(25,32)(26,31);(2,8,12,11)(9,24,13,23)(10,17,28,27)(25,30,29,26);(1,3)(2,9,12,13)(4,18)(5,22)(6,15)(7,16)(8,23,11,24)(10,26,28,30)(14,19)(17,29,27,25)(20,32)(21,31);(1,4,5,19,7,20,6,21)(2,10,11,27,12,28,8,17)(3,14,15,18,16,31,22,32)(9,25,23,26,13,29,24,30)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 375332);