#491400

order	512
nilpotency class	4
isoclinism	isoclinic class of #491227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19128	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

408 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398596);