#199715

order	512
nilpotency class	3
isoclinism	isoclinic class of #199558
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3217	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1235 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106911);