#430410

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1197 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)(5,6,10,16,12,18,11,17)(7,19,9,27,22,15,21,13)(8,14,24,31,26,32,25,33)(20,29,23,36,34,30,35,28);(1,3,4,2)(5,7,12,22)(6,13,18,27)(8,23,26,35)(9,10,21,11)(14,29,32,30)(15,17,19,16)(20,25,34,24)(28,33,36,31);(2,3)(5,8)(6,14)(7,20)(9,23)(10,24)(11,25)(12,26)(13,28)(15,30)(16,31)(17,33)(18,32)(19,29)(21,35)(22,34)(27,36);(2,3)(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,27)(8,23,24,34,26,35,25,20)(14,30,31,28,32,29,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337606);