#391740

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

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

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, 298936);