#217103

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

839 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)(9,10)(11,14)(12,15)(13,16)(17,18,23,26,21,24,22,25)(19,27,30,31,28,20,29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,28)(22,29)(23,30)(24,27)(25,31)(26,32);(2,5)(6,8)(10,15)(11,12)(14,16)(17,20,21,27)(18,19,24,28)(22,32,23,31)(25,30,26,29);(1,3,4,7)(2,6,5,8)(18,24)(19,28)(25,26)(29,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=ehi,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, 124299);