#396836

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

993 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)(6,12)(7,17)(8,22)(9,10)(15,16)(19,33)(20,21)(23,24)(25,28)(26,30)(27,36)(32,35);(5,6,9,12,11,15,10,16)(7,18,17,25,21,13,20,28)(8,14,23,26,24,29,22,30)(19,32,31,35,33,27,34,36);(5,7)(6,13)(8,19)(9,20)(10,17)(11,21)(12,25)(14,27)(15,18)(16,28)(22,31)(23,34)(24,33)(26,35)(29,32)(30,36);(2,4)(5,8,9,23,11,24,10,22)(6,14,12,26,15,29,16,30)(7,19,20,34,21,33,17,31)(13,27,25,35,18,32,28,36)

Polycyclic presentation:

2^9|b^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,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, 304032);