#346989

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2175 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 2175 conjugacy classes of subgroups — too many to draw. Only its 222 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254185);