#406109

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 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:

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

Polycyclic presentation:

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

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, 313305);