#193969

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(3,9,10,16)(4,8)(5,7)(6,11,14,12)(13,15)(18,23,26,25)(19,27,29,28);(1,3,4,12,5,10,13,11)(2,6,8,16,7,14,15,9)(17,18,20,23,22,26,21,25)(19,24,27,30,29,32,28,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32);(2,7)(3,10)(8,15)(11,12)(17,20,22,21)(18,25,26,23)(19,28,29,27)(24,30,32,31)

Polycyclic presentation:

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

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, 101165);