#6788264

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	#31057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 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)(3,6)(4,5)(7,8)(17,18,21,24)(19,26,28,32)(20,23,30,25)(22,27,31,29);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,25)(21,24)(23,30)(26,29)(27,32)(28,31);(2,5)(6,8)(10,13)(14,16)(17,19,20,29,21,28,30,27)(18,22,25,26,24,31,23,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,25)(21,24)(23,30)(26,29)(27,32)(28,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,30)(18,23,24,25)(19,27,28,29)(22,26,31,32)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=h,e^2=f,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, 6695460);