#6463878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(3,6)(4,8)(5,7)(17,18,20,25,21,24,28,22)(19,23,27,31,26,30,32,29);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,30)(25,31)(28,32);(9,11,12,16)(10,14,13,15)(18,24)(22,25)(23,30)(29,31);(19,26)(23,30)(27,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371074);