#427886

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1244 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=ef,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,f^b=fi,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, 335082);