#6494873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,23,28)(19,25,20,31)(21,26,22,27)(24,29,32,30);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,30)(22,29)(25,27)(26,31)(28,32);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,20,23,19)(18,25,28,31)(21,30,22,29)(24,26,32,27);(10,13)(14,16)(17,21,23,22)(18,26,28,27)(19,29,20,30)(24,25,32,31);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,27,28,26)(19,30,20,29)(24,25,32,31)

Polycyclic presentation:

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

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, 6402069);