#193382

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

856 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)(3,7)(4,10)(5,9)(6,11)(8,12)(13,16)(14,15)(17,18,22,26,23,20,21,25)(19,27,30,24,31,32,29,28);(1,3)(2,7)(4,14)(5,12)(6,13)(8,9)(10,15)(11,16)(17,19,22,30,23,31,21,29)(18,24,26,32,20,28,25,27);(1,3)(2,8)(4,12)(5,14)(6,13)(7,9)(10,16)(11,15)(17,20)(18,23)(19,28)(21,26)(22,25)(24,31)(27,30)(29,32);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,15,16,8)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,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, 100578);