#416988

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1044 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)(5,6)(7,8)(9,10,13,14)(11,15,12,16)(17,18)(19,25)(20,26)(21,27)(22,24)(23,29)(28,30)(31,32);(5,6)(7,8)(10,14)(15,16)(17,19,25,18,22,27,21,24)(20,28,32,31,30,26,29,23);(2,4)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,29,22,30,25,32)(18,23,19,26,24,31,27,28);(9,12,13,11)(10,15,14,16)(18,24)(19,27)(23,31)(26,28)

Polycyclic presentation:

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

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, 324184);