#442673

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(7,8)(10,13)(11,15)(17,18,22,27,23,28,39,29)(19,30,32,41,33,26,21,24)(20,34,37,46,38,25,42,45)(31,44,35,40,47,43,36,48);(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,24,28,41)(20,35,38,36)(21,39,32,22)(25,43,34,44)(26,29,30,27)(31,42,47,37)(40,46,48,45);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,22,37,23,38,39,42)(18,25,27,45,28,34,29,46)(19,31,32,35,33,47,21,36)(24,40,30,43,41,48,26,44);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,21,22,19,23,32,39,33)(18,26,27,24,28,30,29,41)(20,36,37,31,38,35,42,47)(25,44,45,40,34,43,46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=efgi,c^2=e,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=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, 349869);