#478916

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

806 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)(5,6,10,16,9,17,11,15)(7,12,26,28,19,14,20,33)(8,21,24,30,25,31,23,13)(18,29,22,27,34,36,35,32);(5,7,11,26,9,19,10,20)(6,12,15,28,17,14,16,33)(8,22,23,34,25,35,24,18)(13,29,31,27,30,36,21,32);(5,8,11,23,9,25,10,24)(6,13,15,31,17,30,16,21)(7,18,26,22,19,34,20,35)(12,27,28,36,14,32,33,29);(2,4)(5,7)(6,14)(8,18)(9,19)(10,26)(11,20)(12,17)(13,27)(15,28)(16,33)(21,36)(22,24)(23,35)(25,34)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,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,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386112);