#193380

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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)(3,6)(4,7)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,22,25,23,20,21,26)(19,27,29,24,31,32,30,28);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,22,29,23,31,21,30)(18,24,25,32,20,28,26,27);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,23)(19,28)(21,25)(22,26)(24,31)(27,29)(30,32);(2,8)(3,10)(7,9)(11,16)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,e^c=eh,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, 100576);