#254134

order	512
nilpotency class	3
isoclinism	isoclinic class of #254131
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1020 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1020 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)(3,7)(4,6)(5,8)(10,12)(16,21);(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,17)(14,16,22,21)(15,23)(18,19)(20,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,18)(13,20)(14,23)(16,24)(17,21)(19,22);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,18,20,15)(14,21,22,16)(17,19,24,23)

Polycyclic presentation:

2^9|b^2=e,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161330);