#441225

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1225 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)(5,6,9,16)(7,12)(8,14)(10,15,11,25)(13,23)(17,33,20,26)(18,28)(19,27)(21,32)(22,30)(24,31)(29,34,36,35);(5,7,10,23,9,18,11,19)(6,12,15,13,16,28,25,27)(8,20,24,35,21,17,22,34)(14,26,31,29,32,33,30,36);(2,4)(5,7,9,18)(6,13)(8,17,21,20)(10,19,11,23)(12,15)(14,29)(16,27)(22,34,24,35)(25,28)(26,31)(30,33)(32,36);(5,8)(6,14,16,32)(7,17)(9,21)(10,24)(11,22)(12,26,28,33)(13,29,27,36)(15,31,25,30)(18,20)(19,35)(23,34)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,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, 348421);