#6435857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(11,15)(14,16)(18,22)(19,25)(20,27)(23,30)(24,29)(26,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22,21,24,27,29)(19,23,25,28,26,30,31,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,20,25,21,26,27,31)(18,23,22,28,24,30,29,32);(9,11,12,15)(10,14,13,16);(18,24)(22,29)(23,30)(28,32)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=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, 6343053);