#299731

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 915 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=eh,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, 206927);