#102750

order	512
nilpotency class	3
isoclinism	isoclinic class of #102749
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	#28381	C₂
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

606 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,f^b=fghi,d^c=dh,e^c=ei,f^c=fghi

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, 9946);