#201088

order	512
nilpotency class	3
isoclinism	isoclinic class of #201085
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1309 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)(3,5)(4,12,14,8)(6,11)(7,9)(10,13,16,15)(17,18)(19,22,26,30)(20,23)(21,24)(25,29,27,31)(28,32);(1,3,5,2,7,11,6,9)(4,10,13,8,14,16,15,12)(17,19,21,26)(18,22)(20,27,28,25)(23,31)(24,30)(29,32);(1,4)(2,8)(3,10)(5,13)(6,15)(7,14)(9,12)(11,16);(1,5,7,6)(2,3,9,11)(4,13,14,15)(8,10,12,16)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108284);