#401260

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1243 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308456);