#507519

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eghi,b^2=egi,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414715);