#416974

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi

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, 324170);