#192966

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

888 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100162);