#479890

order	512
nilpotency class	3
isoclinism	isoclinic class of #479576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=g,d^2=efg,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi

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, 387086);