#146274

order	512
nilpotency class	4
isoclinism	isoclinic class of #146231
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=dg,b^2=degh,c^2=i,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=dgi,e^c=egi,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224920, #72224925, #72224926
maximal normal subgroup of	#72224920, #72224925, #72224926

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53470);