#6785395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#38050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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)(17,18)(19,25)(20,22)(21,26)(23,30)(24,27)(28,32)(29,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,24,28)(18,25,27,32)(20,29,30,26)(21,22,31,23);(3,8)(5,7)(10,13)(14,16)(17,20,22,18,24,30,23,27)(19,26,31,32,28,29,21,25);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,22)(20,25)(23,28)(24,31)(27,29)(30,32);(10,13)(14,16)(17,22,24,23)(18,20,27,30)(19,21,28,31)(25,26,32,29)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6692591);