#195784

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

1082 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,e^2=h,f^2=hi,g^2=i,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, 102980);