#397730

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1239 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304926);