#382556

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,h^b=hi,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289752);