#303921

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45996	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1035 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1035 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,21)(19,20)(23,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,39)(38,40);(1,3,5,4)(2,6,8,7)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,20)(23,24)(27,34)(28,37)(31,36)(33,39);(1,3,5,4)(2,7,8,6)(9,11,12,21,14,20,13,23)(10,15,16,18,17,24,22,19)(25,27,28,35,29,34,37,40)(26,30,31,33,32,38,36,39);(1,4,5,3)(2,7,8,6)(11,20)(15,24)(18,19)(21,23)

Polycyclic presentation:

2^9|b^2=e,c^2=ef,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211117);