#415516

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×C₄³ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322712);