#465847

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

1029 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 373043);