#507741

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 122 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)(11,21)(12,26)(13,19)(14,17)(15,20)(16,22)(18,23)(24,32)(25,33)(27,36)(28,34)(29,39)(30,40)(31,35)(37,38);(1,3,4,8)(2,5,6,7)(9,11,13,14,16,24,15,25)(10,17,20,21,22,33,19,32)(12,27,28,38,30,31,29,23)(18,34,35,26,37,39,36,40);(9,12,16,30)(10,18,22,37)(11,23,24,38)(13,29,15,28)(14,31,25,27)(17,26,33,40)(19,35,20,36)(21,34,32,39);(2,6)(5,7)(10,19,22,20)(11,24)(14,25)(17,21,33,32)(18,35,37,36)(23,38)(26,34,40,39)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414937);