#499691

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=efh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406887);