#465091

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

929 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=e,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,d^c=de,e^c=ei,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, 372287);