#434080

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1184 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:

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

Polycyclic presentation:

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

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, 341276);