#401881

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 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)(12,23)(13,26)(17,33)(18,29)(20,36)(21,22)(24,25)(27,31)(30,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,18,26,29)(11,16,21,27,22,32,19,31)(12,17,23,28,25,33,24,34)(20,30,37,38,36,40,35,39);(3,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,27)(17,30)(18,32)(19,26)(23,37)(24,35)(25,36)(28,38)(29,31)(33,40)(34,39);(9,12,13,23,14,25,26,24)(10,17,15,28,18,33,29,34)(11,20,21,37,22,36,19,35)(16,30,27,38,32,40,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=e,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,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, 309077);