#6463879

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

1478 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)(17,18,20,24,21,23,29,28)(19,22,27,30,26,25,31,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,29)(27,30)(31,32);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31);(9,11,12,16)(10,14,13,15)(18,23)(22,25)(24,28)(30,32);(19,26)(22,25)(27,31)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=hi,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, 6371075);