#256484

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163680);