#299726

order	512
nilpotency class	3
isoclinism	isoclinic class of #299717
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18714	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 966 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=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, 206922);