#98123

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2682	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

534 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,7,4,3,8,6)(9,10)(11,18,20,15)(12,16)(13,17)(14,19,23,21)(22,24)(25,26,29,32,30,27,33,35)(28,31,36,39,37,34,38,40);(5,8)(6,7)(9,11)(10,14)(12,21)(13,20)(15,16)(17,23)(18,24)(19,22)(25,27)(26,30)(28,34)(29,32)(31,37)(33,35)(36,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,11)(10,15)(12,19)(13,20)(14,16)(17,18)(21,22)(23,24)(25,28,30,37)(26,31,27,34)(29,38,33,36)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5319);