#192894

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 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:

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

Polycyclic presentation:

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

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, 100090);