#350109

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

828 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,20,15,16)(12,22,13,24)(17,21,18,23)(25,26,30,29)(27,34,31,32)(28,36,38,40)(33,39,37,35);(2,5)(6,8)(9,11,14,15)(10,16,19,20)(12,21,13,23)(17,24,18,22)(25,27,30,31)(26,32,29,34)(28,35,38,39)(33,36,37,40);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,19,18)(11,21,15,23)(16,24,20,22)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,36)(34,40);(9,13,14,12)(10,18,19,17)(11,21,15,23)(16,24,20,22)(25,29,30,26)(27,32,31,34)(28,37,38,33)(35,36,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257305);