#476843

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 384039);