#219275

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126471);