#6815345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
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	#3045	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,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,d^c=di,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722541);