#401661

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(13,24)(16,28)(17,26)(18,27)(22,35)(25,30)(29,36)(31,37)(34,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,18,24,27)(11,16,21,25,19,30,33,28)(12,17,22,26,23,31,35,37)(20,29,34,36,32,39,40,38);(3,7)(5,8)(9,11,13,21,14,19,24,33)(10,16,15,25,18,30,27,28)(12,20,22,34,23,32,35,40)(17,29,26,36,31,39,37,38);(9,12)(10,17)(11,20)(13,22)(14,23)(15,26)(16,29)(18,31)(19,32)(21,34)(24,35)(25,36)(27,37)(28,38)(30,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 308857);