#6815337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31228	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722533);