#495956

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=gi,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=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, 403152);