#427996

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1276 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)(5,6)(7,8)(9,10,14,20,16,22,15,21)(11,23,25,18,12,27,26,17)(13,29,31,19,32,37,33,38)(24,35,30,39,28,34,40,36);(2,3)(9,11,16,12)(10,17,22,18)(13,30,32,40)(14,26,15,25)(19,36,38,39)(20,27,21,23)(24,33,28,31)(29,35,37,34);(9,12,16,11)(10,18,22,17)(13,28,32,24)(14,25,15,26)(19,35,38,34)(20,23,21,27)(29,39,37,36)(30,33,40,31);(1,3,4,2)(6,8)(9,13,16,32)(10,19,22,38)(11,24,12,28)(14,33,15,31)(17,34,18,35)(20,29,21,37)(23,39,27,36)(25,40,26,30)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 335192);