#467053

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 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)(6,12)(7,17)(9,10)(14,30)(15,16)(18,28)(19,22)(20,21)(23,24)(26,32)(31,36)(33,34);(5,6,10,15,11,16,9,12)(7,18,20,29,21,28,17,13)(8,14,23,30,25,32,24,26)(19,31,22,35,34,36,33,27);(2,4)(5,7,11,21)(6,13,16,29)(8,22,25,33)(9,20,10,17)(12,18,15,28)(14,31,32,36)(19,24,34,23)(26,35,30,27);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,26)(13,27)(15,30)(16,32)(17,33)(18,31)(20,22)(21,34)(28,36)(29,35)

Polycyclic presentation:

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

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, 374249);