#476656

order	512
nilpotency class	3
isoclinism	isoclinic class of #476654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

1048 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)(7,17)(8,22,25,14)(9,15)(10,13)(11,16)(12,19)(18,34,23,33)(20,29)(21,27)(24,30,26,32)(28,35,31,36);(1,2)(3,4)(5,7,11,21)(6,12,16,29)(8,23,25,18)(9,20,10,19)(13,27,15,17)(14,31,22,28)(24,35,26,36)(30,33,32,34);(2,4)(6,13)(8,24)(9,10)(12,27)(14,22)(15,16)(17,29)(18,35)(19,20)(23,36)(25,26)(33,34);(5,8,9,24,11,25,10,26)(6,14,13,30,16,22,15,32)(7,18,19,35,21,23,20,36)(12,28,17,33,29,31,27,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383852);