#246990

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60645	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1359 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1359 conjugacy classes of subgroups — too many to draw. Only its 150 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)(11,23)(12,28)(13,30)(14,21)(15,20)(16,22)(17,25)(18,29)(19,31)(24,36)(26,38)(27,37)(32,35)(33,40)(34,39);(3,7)(5,8)(9,11,16,27)(10,17,22,38)(12,13,24,32)(14,26,15,25)(18,30,39,35)(19,36,40,28)(20,37,21,23)(29,33,34,31);(1,2)(3,8)(4,6)(5,7)(9,12,16,24)(10,18,22,39)(11,13,27,32)(14,29,15,34)(17,35,38,30)(19,37,40,23)(20,28,21,36)(25,31,26,33);(3,7)(5,8)(9,13)(10,19)(11,24)(12,27)(14,33)(15,31)(16,32)(17,36)(18,37)(20,35)(21,30)(22,40)(23,39)(25,29)(26,34)(28,38)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154186);