#346559

order	512
nilpotency class	3
isoclinism	isoclinic class of #346550
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62894	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2413 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 2413 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253755);