#6519142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37550	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6426338);