#401599

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 308795);