#160653

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

926 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67849);