#458029

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 365225);