#476614

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1004 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:

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

Polycyclic presentation:

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

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, 383810);