#455689

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 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)(7,15)(8,19)(9,23)(10,14)(11,17)(12,21)(13,26)(16,33)(18,28)(20,36)(22,31)(24,30)(25,29)(27,35)(32,34);(5,7,9,24,10,17,26,18)(6,11,13,30,14,15,23,28)(8,20,21,34,22,16,25,35)(12,27,19,33,29,32,31,36);(5,8,10,22)(6,12,14,29)(7,16,17,20)(9,25,26,21)(11,27,15,32)(13,31,23,19)(18,35,24,34)(28,33,30,36);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,17)(16,32)(18,28)(19,21)(20,27)(22,29)(23,26)(24,30)(25,31)(33,34)(35,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=i,f^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,g^b=gi,d^c=de,f^c=fi,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, 362885);