#324465

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,19,23)(12,15,20,24)(17,21,18,22)(25,26)(27,32)(28,29)(30,33)(31,34)(35,36)(37,39)(38,40);(2,5)(6,7)(9,11,12,18,13,19,20,17)(10,14,15,22,16,23,24,21)(25,27,30,38,31,36,29,37)(26,32,33,40,34,35,28,39);(11,17)(12,20)(14,21)(15,24)(18,19)(22,23)(25,28,30,34,31,33,29,26)(27,35,38,40,36,32,37,39);(1,3,4,8)(2,6,5,7)(27,36)(32,35)(37,38)(39,40)

Polycyclic presentation:

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

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, 231661);