#199805

order	512
nilpotency class	3
isoclinism	isoclinic class of #199797
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1252 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)(4,11)(6,9)(7,13)(8,10)(12,16)(14,15)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,3,10,9,6,8,5,2)(4,11,13,16,14,15,12,7)(17,19,21,26)(18,22)(20,25,27,31)(23,28)(24,29)(30,32);(1,4,6,14)(2,7,9,16)(3,11,8,15)(5,12,10,13)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32);(1,5,6,10)(2,3,9,8)(4,12,14,13)(7,11,16,15)(19,26)(22,29)(25,31)(28,32)

Polycyclic presentation:

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

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, 107001);