#254266

order	512
nilpotency class	3
isoclinism	isoclinic class of #254227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24222	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 915 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

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, 161462);