#464022

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371218);