#97510

order	512
nilpotency class	3
isoclinism	isoclinic class of #97486
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

560 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4706);