#472278

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

954 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)(7,13)(8,21)(9,12)(10,16)(11,15)(14,24)(17,26)(18,31)(19,29)(20,30)(22,28)(23,32)(25,27)(33,36)(34,35);(1,2,3,4)(6,12)(7,17)(9,11)(14,27)(15,16)(18,22)(19,20)(21,32)(23,24)(26,29)(31,36)(33,34);(2,4)(5,7,11,19,10,20,9,17)(6,13,12,26,16,30,15,29)(8,22,23,18,25,34,24,33)(14,31,27,28,32,36,21,35);(5,8)(6,14)(7,18)(9,24)(10,25)(11,23)(12,27)(13,28)(15,21)(16,32)(17,22)(19,34)(20,33)(26,36)(29,31)(30,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379474);