#465211

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 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)(7,17)(8,23)(9,15)(10,14)(11,16)(12,20)(13,26)(18,21)(19,28)(22,29)(24,27)(25,30)(31,35)(32,33)(34,36);(5,7)(6,12,16,29)(8,19)(9,20)(10,22)(11,21)(13,28,30,36)(14,17,15,18)(23,32,24,31)(25,33)(26,35)(27,34);(5,6)(7,18,21,17)(8,24)(9,15)(10,14)(11,16)(12,20,29,22)(13,25)(19,28,34,36)(23,27)(26,30)(31,33,32,35);(2,4)(5,8,9,25,11,27,10,26)(6,13,14,24,16,30,15,23)(7,19,20,33,21,34,22,35)(12,28,17,31,29,36,18,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372407);