#364444

order	512
nilpotency class	3
isoclinism	isoclinic class of #196344
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35149	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2460 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)(7,12,20,29,19,28,18,27)(8,21,23,32,24,13,25,31)(17,30,33,36,22,26,34,35);(2,4)(5,7,10,20,11,19,9,18)(6,12,16,29,15,28,14,27)(8,22,23,34,24,17,25,33)(13,30,31,36,21,26,32,35);(5,8)(6,13)(7,17)(9,23)(10,25)(11,24)(12,26)(14,31)(15,21)(16,32)(18,33)(19,22)(20,34)(27,35)(28,30)(29,36);(2,4)(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,23,24,25)(12,27,28,29)(13,31,21,32)(17,33,22,34)(26,35,30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,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, 271640);