#6786411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#38024	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

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, 6693607);