#237934

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

842 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145130);