#96784

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cgi,d^b=dh

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, 3980);