#391599

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 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)(5,6,10,16,11,12,9,15)(7,17,20,13,21,29,18,26)(8,14,25,31,24,27,22,30)(19,32,34,28,23,35,33,36);(6,12)(7,18)(8,22)(9,10)(13,17)(14,30)(19,23)(20,21)(24,25)(26,29)(27,31)(32,35);(5,7,11,21)(6,13,12,26)(8,23,24,19)(9,20,10,18)(14,28,27,36)(15,29,16,17)(22,33,25,34)(30,35,31,32);(5,8,9,22,11,24,10,25)(6,14,15,30,12,27,16,31)(7,19,18,33,21,23,20,34)(13,28,17,32,26,36,29,35)

Polycyclic presentation:

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

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, 298795);