#326565

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2580
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

439 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307163, #72307201, #72307233
maximal normal subgroup of	#72307163, #72307201, #72307233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233761);