#394605

order	512
nilpotency class	3
isoclinism	isoclinic class of #394602
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2195 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,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, 301801);