#324605

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
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 150 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 150 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)(9,10,14,17)(11,15,19,22)(12,21,16,13)(18,24,23,20)(25,26,28,31)(27,33,36,30)(29,32,34,38)(35,40,37,39);(2,5)(6,7)(9,11)(10,15)(12,18)(13,20)(14,19)(16,23)(17,22)(21,24)(25,27,34,35,28,36,29,37)(26,30,38,39,31,33,32,40);(9,12,13,10,14,16,21,17)(11,18,20,15,19,23,24,22)(27,35)(29,34)(30,39)(32,38)(33,40)(36,37);(1,3,4,8)(2,6,5,7)(11,19)(15,22)(18,23)(20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231801);