#217095

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 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)(3,7)(10,13)(11,14)(15,16)(17,18,23,26,22,27,21,30)(19,25,28,31,20,24,29,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,20)(18,24,27,25)(21,29,23,28)(26,32,30,31);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,25,27,24)(21,29,23,28)(26,32,30,31);(1,3,4,7)(2,6,5,8)(9,10,12,13)(11,14,15,16)(17,21,22,23)(18,26,27,30)(19,28,20,29)(24,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124291);