#490679

order	512
nilpotency class	4
isoclinism	isoclinic class of #490472
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

422 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)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,28)(20,25)(21,34)(22,37)(23,27)(24,32)(26,35)(29,42)(30,43)(31,44)(33,45)(36,38)(39,47)(40,48)(41,46);(17,19,29,46,22,31,21,32,23,30,35,48,20,33,36,47)(18,24,38,44,25,41,26,28,27,39,34,45,37,40,42,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(19,30)(24,39)(28,43)(31,33)(32,47)(40,41)(44,45)(46,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,22)(18,25,27,37)(19,31,30,33)(21,29,36,35)(24,40,39,41)(26,34,42,38)(28,44,43,45)(32,48,47,46)

Polycyclic presentation:

2^9|b^2=egi,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=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287072, #72287105, #72287115, …
maximal normal subgroup of	#72287072, #72287105, #72287115, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397875);