#455683

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(6,12)(7,16)(9,10)(13,27)(14,26)(17,25)(18,32)(19,20)(21,24)(28,36)(29,33)(34,35);(5,6,9,23,11,12,10,15)(7,17,16,13,20,27,19,25)(8,14,21,31,22,26,24,30)(18,33,32,28,34,36,35,29);(5,7,11,20)(6,13,12,25)(8,18,22,34)(9,19,10,16)(14,29,26,28)(15,27,23,17)(21,35,24,32)(30,33,31,36);(2,4)(5,8)(6,14)(7,18)(9,21)(10,24)(11,22)(12,26)(13,28)(15,30)(16,32)(17,33)(19,35)(20,34)(23,31)(25,29)(27,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,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, 362879);