#205798

order	512
nilpotency class	3
isoclinism	isoclinic class of #205795
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

980 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268383, #72268392, #72268408, …
maximal normal subgroup of	#72268383, #72268392, #72268408, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112994);