#471071

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16742
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

641 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:

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

Polycyclic presentation:

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

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, 378267);