#364431

order	512
nilpotency class	3
isoclinism	isoclinic class of #196161
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2416 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)(5,6,10,16,11,15,9,14)(7,12,19,28,20,29,18,27)(8,21,23,13,25,30,24,31)(17,32,33,36,22,35,34,26);(2,4)(5,7,10,19,11,20,9,18)(6,12,16,28,15,29,14,27)(8,22,23,34,25,17,24,33)(13,26,30,32,31,36,21,35);(5,8)(6,13)(7,17)(9,23)(10,24)(11,25)(12,26)(14,30)(15,31)(16,21)(18,33)(19,34)(20,22)(27,32)(28,35)(29,36);(2,4)(5,9,11,10)(6,14,15,16)(7,18,20,19)(8,23,25,24)(12,27,29,28)(13,30,31,21)(17,33,22,34)(26,32,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271627);