#294982

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,19,22)(12,21,16,13)(18,24,23,20)(25,26,30,33)(27,31,34,39)(28,36,35,40)(29,38,37,32);(2,5)(6,7)(9,11,14,19)(10,15,17,22)(12,18,16,23)(13,20,21,24)(25,27,30,34)(26,31,33,39)(28,37,35,29)(32,40,38,36);(9,12,13,10,14,16,21,17)(11,18,20,15,19,23,24,22)(25,28,29,27,30,35,37,34)(26,31,32,36,33,39,38,40);(1,3,4,8)(2,6,5,7)(11,19)(15,22)(18,23)(20,24)(27,34)(28,35)(31,39)(36,40)

Polycyclic presentation:

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

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, 202178);