#219359

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32)(33,34,38,41,39,42,37,46)(35,40,36,45,44,47,43,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,18,21,23)(19,22,27,30)(20,24,26,29)(25,28,31,32)(33,35,37,43,39,44,38,36)(34,40,46,48,42,47,41,45);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28)(33,36,38,44,39,43,37,35)(34,40,46,48,42,47,41,45);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(18,23)(19,27)(24,29)(25,31)

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=ce,d^b=dhi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126555);