#219216

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 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)(11,15)(14,16)(17,18,23,19,22,25,21,27)(20,24,29,26,30,32,31,28);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,19,23,25,22,27,21,18)(20,28,29,24,30,26,31,32);(2,5)(6,8)(9,11,13,15,12,16,10,14)(17,20)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28);(1,3,4,7)(2,6,5,8)(9,10,12,13)(11,15,16,14)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,28,32,26)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,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, 126412);