#416142

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	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1680 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eh,c^2=i,d^2=h,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, 323338);