#413625

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
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

856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320821);