#422651

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1081 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329847);