#404591

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311787);