#465832

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 373028);