#450829

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

970 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)(17,18,22,27,21,26,37,45)(19,23,32,42,30,40,31,41)(20,33,36,44,34,24,35,25)(28,39,29,47,46,38,48,43);(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,30)(18,23,26,40)(20,29,34,48)(22,32,37,31)(24,43,33,47)(25,39,44,38)(27,42,45,41)(28,36,46,35);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,28)(21,34)(22,36)(23,38)(25,27)(26,33)(29,32)(30,46)(31,48)(35,37)(39,40)(41,47)(42,43)(44,45);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,39)(24,27)(26,44)(28,32)(30,48)(31,46)(33,45)(36,37)(38,40)(41,47)(42,43)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 358025);