#146032

order	512
nilpotency class	4
isoclinism	isoclinic class of #146013
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#470
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 204 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) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,e^c=eg,f^c=fhi,g^c=gi

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, 53228);