#427840

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 335036);