#501697

order	512
nilpotency class	4
isoclinism	isoclinic class of #501695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408893);