#160628

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1138 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh

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, 67824);