#394636

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2266 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:

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

Polycyclic presentation:

2^9|c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,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, 301832);