#375672

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282868);