#491542

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=gi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg

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, 398738);