#205155

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

896 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112351);