#161316

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

834 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)(9,10,13,15,14,11,12,16)(17,18)(19,23)(20,22)(21,24)(25,28)(26,31)(27,29)(30,32);(9,11)(10,14)(12,16)(13,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(9,12,14,13)(10,15,11,16)(18,24)(19,26)(22,25)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68512);