#198409

order	512
nilpotency class	3
isoclinism	isoclinic class of #198126
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1200 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)(3,7,12,15)(4,8)(5,13)(6,9)(10,14,11,16)(17,18,21,23,22,25,20,26)(19,27,29,24,30,32,28,31);(1,3)(2,7,9,15)(4,10)(5,11)(6,12)(8,14,13,16)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,19,22,30)(18,24,25,31)(20,29,21,28)(23,27,26,32);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(18,25)(23,26)(24,31)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,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, 105605);