#244253

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=hi,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151449);