#419208

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3012	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1930 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:

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

Polycyclic presentation:

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

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, 326404);