#438219

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 345415);