#6700487

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#25159	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,27,30,24)(20,25,21,26)(28,29,32,31);(1,3,4,8)(2,5,6,7)(17,19,22,30)(18,24,23,27)(20,29,21,31)(25,28,26,32);(2,6)(5,7)(10,13)(14,16)(17,20)(18,25)(19,28)(21,22)(23,26)(24,31)(27,29)(30,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,29)(21,22)(23,25)(24,32)(27,28)(30,31);(3,8)(5,7)(17,21,22,20)(18,26,23,25)(19,28,30,32)(24,31,27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607683);