#408151

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

920 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:

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

Polycyclic presentation:

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

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, 315347);