#224203

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fh,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131399);