#205631

order	512
nilpotency class	3
isoclinism	isoclinic class of #205553
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60940	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1039 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:

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

Polycyclic presentation:

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

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, 112827);