#192582

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

876 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99778);