#507996

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

431 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(9,10,15,17,14,18,22,24,16,12,23,20,13,19,11,21);(1,2,4,5)(3,6,7,8)(9,11,13,23,16,22,14,15)(10,17,18,24,12,20,19,21);(2,5)(6,8)(9,12)(10,16)(11,17)(13,19)(14,18)(15,21)(20,22)(23,24);(1,3,4,7)(2,6,5,8)(9,13,16,14)(10,18,12,19)(11,15,22,23)(17,21,20,24)

Polycyclic presentation:

2^9|a^2=efh,b^2=eg,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415192);