#200019

order	512
nilpotency class	3
isoclinism	isoclinic class of #199851
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	#60798	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1233 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:

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

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=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

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, 107215);