#161900

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#121	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69096);