#395557

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,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, 302753);