#102377

order	512
nilpotency class	3
isoclinism	isoclinic class of #102154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
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

332 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egh,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fh

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, 9573);