#443511

order	512
nilpotency class	3
isoclinism	isoclinic class of #420407
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1190 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) = 128

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

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

Polycyclic presentation:

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

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, 350707);