#195501

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
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

1117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102697);