#357206

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 conjugacy classes of subgroups; only the 378 normal subgroups are shown.

normal subgroups only

This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 378 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264402);