#6435918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
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	C₂×#2657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 538 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=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, 6343114);