#421908

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

1178 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^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=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,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, 329104);