#460305

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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)(5,6,9,14,10,13,25,30)(7,15,18,31,19,11,23,28)(8,20,21,32,22,12,24,29)(16,26,17,27,35,33,36,34);(5,7,10,19)(6,11,13,15)(8,17,22,36)(9,23,25,18)(12,27,20,34)(14,31,30,28)(16,24,35,21)(26,32,33,29);(2,4)(5,8)(6,12)(7,16)(9,24)(10,22)(11,26)(13,20)(14,32)(15,33)(17,23)(18,36)(19,35)(21,25)(27,31)(28,34)(29,30);(5,8)(6,12,13,20)(7,17)(9,21)(10,22)(11,27,15,34)(14,29,30,32)(16,23)(18,35)(19,36)(24,25)(26,28,33,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367501);