#382561

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1986 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)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32);(5,6)(7,8)(10,13)(14,16)(17,19,22,28)(18,23,25,30)(20,29,27,21)(24,32,31,26);(2,4)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,19,22,27,29,28)(18,24,26,23,25,31,32,30);(11,15)(14,16)(18,25)(23,30)(24,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289757);