#507920

order	512
nilpotency class	4
isoclinism	isoclinic class of #501527
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	C₂²×D₈	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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=gh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,d^c=dh,g^c=gi,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, 415116);