#345812

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,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, 253008);