#161790

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(33,34,37,41,38,42,36,40)(35,39,43,46,44,47,45,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25)(33,35)(34,39)(36,45)(37,43)(38,44)(40,48)(41,46)(42,47);(18,23)(22,31)(24,28)(25,32)(33,36,38,37)(34,40,42,41)(35,43,44,45)(39,46,47,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,23)(22,31)(24,28)(25,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,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, 68986);