#395748

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
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	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2138 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)(17,18,22,27,21,26,20,25)(19,28,29,32,30,23,31,24);(1,2)(3,4)(10,13)(14,16)(17,19)(18,23)(20,31)(21,30)(22,29)(24,27)(25,32)(26,28);(2,4)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,28)(26,32);(6,8)(17,20,21,22)(18,25,26,27)(19,29,30,31)(23,24,28,32)

Polycyclic presentation:

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

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, 302944);