#394504

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2229 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)(5,6)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,28)(19,36)(20,34)(21,31)(23,35)(26,30)(27,33)(29,38)(32,39)(37,40);(1,2)(3,4)(9,11,14,21)(10,15,18,31)(12,19,23,37)(13,26,28,20)(16,32,33,29)(17,34,25,30)(22,36,35,40)(24,39,27,38);(1,3,4,2)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,35)(18,33)(20,39)(21,37)(22,25)(24,28)(26,38)(30,40)(31,32)(34,36);(6,8)(12,23)(16,33)(19,37)(22,35)(24,27)(29,32)(36,40)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=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, 301700);