#98832

order	512
nilpotency class	3
isoclinism	isoclinic class of #98828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

522 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cgi,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6028);