#6458357

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,27,24,29,23,28)(19,30,20,25,31,32,21,26);(2,6)(7,8)(17,19)(18,25)(20,23)(21,22)(24,31)(26,29)(27,30)(28,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,19,24,21,22,31)(18,25,28,30,29,26,27,32);(1,4,5,3)(2,8,6,7)(17,21,23,31,24,20,22,19)(18,26,28,32,29,25,27,30);(17,22,24,23)(18,27,29,28)(19,21,31,20)(25,30,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365553);